JERSEY CITY, N.J.—Katie Brennan spent more than a year trying to get authorities to take action against the man she accuses of sexually assaulting her. Finally, she emailed New Jersey Gov. Phil Murphy.

Ms. Brennan, chief of staff at the state’s housing agency and a former volunteer for the governor’s campaign, first called the police in April 2017, one day, she said, after a campaign staffer allegedly forced himself on her. After months of investigation, the county prosecutor’s office declined to file charges against her alleged attacker, saying it didn’t think a jury would convict him. She turned to state law enforcement and high-ranking members of the Murphy administration. Nothing changed.

Finally, the 31-year-old Jersey City resident, who said she turned down a $15,000 settlement offer from her alleged assailant if she signed a nondisclosure agreement, emailed Mr. Murphy and first lady Tammy Murphy directly.

She didn’t explicitly mention the alleged assault in her June 2018 email, but asked to speak with the governor or his wife about a “sensitive matter” that had happened during the campaign. The governor responded within the hour and said he was looping in staff to arrange a time.

“Hang in,” the governor wrote to Ms. Brennan in the email, which was reviewed by The Wall Street Journal. “We are on it.”

A meeting with the governor was never scheduled. And months later, the man, Albert J. Alvarez, was still working for the state, as chief of staff at the New Jersey Schools Development Authority. Mr. Alvarez resigned his position Oct. 2, the same day the Journal emailed him for comment.

“At each turn, I’ve just felt so disappointed,” Ms. Brennan said in an interview. “I tried everything. And none of it worked. If I can’t get any justice, I just don’t seriously know who can.”

Mr. Alvarez’s attorney, John Hogan, said his client “absolutely, positively denies these allegations of sexual assault.” He declined to comment further on any aspect of Ms. Brennan’s account. Neither Mr. Alvarez nor his attorney responded to additional attempts to reach them by phone or to an email with detailed questions.

After his resignation, Mr. Alvarez was named in local and national press reports about the allegation. Ms. Brennan spoke publicly about the alleged assault for the first time for this article.

New Jersey Republicans have called for an investigation into Mr. Alvarez’s hiring, and the state Senate’s Democratic leadership said in a statement they were disturbed by the allegations.

On Friday, Gov. Murphy’s spokesman said his office received a separate accusation on Thursday against Mr. Alvarez. The accusation, which was referred to the New Jersey attorney general’s office, concerned an incident that allegedly occurred in 1999 or 2000, the spokesman said. Neither Mr. Alvarez nor his attorney responded to a request for comment on the allegation.

For all the attention given to allegations of sexual assault in recent months, they remain difficult for the criminal-justice system. Such crimes often produce no conclusive evidence, making them hard to investigate and prosecute. Conflicting accounts can be the only material officials have to work with. Prosecutors tend to decline to pursue cases deemed unlikely to win a conviction.

That can leave accusers feeling they are denied justice. The accused, meanwhile, can still experience life-altering impacts from the allegations alone, including a damaged career and stained reputation.

Criminal-justice officials are now considering new ideas about how to handle such cases. More than a dozen states have passed legislation in the past two years aimed at improving the handling and testing of rape kits, according to the National Conference of State Legislatures. Proposals to broaden the legal definition of sex crimes, and to lengthen or eliminate the statute of limitations for sexual assault have passed or are pending in legislatures across the country.

New training programs for police and prosecutors working on assault cases also are being rolled out, including in Alaska and Illinois.

The governor’s spokesman said the Murphys didn’t find out that Ms. Brennan had accused Mr. Alvarez of sexual assault until the first week of October, when the Journal contacted the administration for comment. In a joint statement, Gov. Murphy and Ms. Murphy said they had asked the director of the Statewide Division of Equal Employment Opportunity and Affirmative Action to review how the state handles allegations of sexual misconduct.

“We are confident that this allegation was handled appropriately by the administration and that policies and procedures were properly and promptly followed. However, it is clear that the process during the transition was inconsistent with our values, and the hire should not have happened,” the Murphys said. “We must now ask: how can we hold ourselves to a higher standard moving forward?”

John L. Molinelli, a former prosecutor of Bergen County, N.J., speaking generally, said sex-crime cases are notoriously difficult to take to trial. Photo: Kevin R. Wexler/The Record of Bergen County/Associated Press

John L. Molinelli, a former prosecutor of Bergen County, N.J., speaking generally, said sex-crime cases are notoriously difficult to take to trial. In order to move forward with charges, prosecutors must reasonably believe that they can prove the case in court.

“Even if they believe he did it,” Mr. Molinelli said, “do they have the evidence that any reasonable prosecutor can believe would sustain a prosecution?”

Helpful information includes the victim confiding in a third party soon after the alleged assault, other victims coming forward, inconsistent statements from the defendant or physical corroboration such as bruises or scratches, Mr. Molinelli said.

Ms. Brennan reported her assault to authorities, went to the hospital for a sexual-assault examination and immediately told friends and family of the alleged attack.

By last fall, Ms. Brennan said she felt confident that the Hudson County prosecutor’s office would arrest Mr. Alvarez. The #MeToo movement had just taken off, prompting many women to step forward with previously unreported stories of sexual assault and harassment, some dating back decades.

She said she was crushed when she found out in early December that county prosecutors wouldn’t move forward. “I closed my office door and I just bawled,” she said. “I felt like I wasn’t heard.”

Ms. Brennan said Assistant Prosecutor Jane Weiner told her a sexual-assault examination performed two days after the alleged attack found saliva and DNA but it was deemed not strong enough. Ms. Brennan said prosecutors also said the alleged assailant claimed the encounter was consensual.

A spokesman for the Hudson County prosecutor’s office said the office doesn’t comment on its investigations.

The special-victims unit in the Hudson County prosecutor’s office investigated 365 cases last year, including adult-sex crimes, child-sexual abuse, physical-abuse and others, according to a spokesman. Of those, prosecutors charged 81 defendants, which led to 11 defendants immediately pleading guilty and 65 indictments. It isn’t clear how the 65 indictments were resolved. The spokesman said those statistics weren’t readily available.

The New Jersey capitol building in Trenton. Photo: Seth Wenig/Associated Press

Trauma, fear of physical or professional retaliation and worries that they won’t be believed prevent many people from reporting their assaults to law enforcement, experts say.

There are no national statistics on the number of sexual-assault cases that end in arrests, prosecution or conviction, said Cassia Spohn, director of the School of Criminology and Criminal Justice at Arizona State University. Federal statistics gathered from local police departments don’t detail how many cases are brought to prosecutors, she said.

The advent of DNA testing has made it harder for defendants to claim sexual encounters never happened. Now, the most common defense in sexual-assault cases is saying the encounter was consensual, said Patti Powers, who served as a senior deputy prosecuting attorney in Washington state for 27 years and who now works for AEquitas, a nonprofit organization that provides training on effective prosecution practices for sex-crime cases.

Oklahoma’s legislature rewrote the state’s rape statute in 2016 after a court ruled that an intoxicated and unconscious teenage victim couldn’t have been forced to perform oral sex, said Ben Fu, a former special-victims unit director for the Tulsa County district attorney’s office. Under the prior law, performing certain sex acts on intoxicated people didn’t constitute forcing them to do those acts.

“We would have jurors go out of courtrooms not at all disputing what happened, but arguing about whether it met their personal definition of force,” Mr. Fu said, referring to sexual-assault cases.

In the wake of the sexual-assault charges against former Michigan State University sports-medicine doctor Larry Nassar, Michigan state legislators moved forward with a raft of bills on sexual assault, including one that would allow prosecutors to use accusations from other victims as corroborating evidence in court. The measure passed the state House of Representatives and is awaiting final vote in the Senate.

Baltimore City State’s Attorney Marilyn J. Mosby said she and other advocates spent five years lobbying for new legislation before the #MeToo movement helped get it passed. Photo: Jerry Jackson/TNS/ZUMA PRESS

In Maryland, a similar law took effect in July. It allows prosecutors in certain circumstances to use prior accusations to counter a defendant’s claim that an encounter was consensual. Marilyn J. Mosby, state’s attorney for Baltimore City, said she and other advocates spent five years lobbying for the legislation before the #MeToo movement helped get it passed. Ms. Mosby said her office is already planning to use the provisions of the new law in four pending cases.

In Illinois, a law requires police officers to undergo at least 40 hours of mandatory sensitivity training every five years to help them better investigate and interview sexual-assault victims.

Ms. Brennan said she realized sexual-assault cases are difficult to win. “But I wanted to try anyway,” she said. “I wanted to have my day in court.”

In April 2017, after a gathering of Murphy campaign staffers at a bar in Jersey City, Mr. Alvarez, who was then the campaign’s outreach director for Muslim and Latino communities, offered to drive Ms. Brennan home and then asked to use her bathroom and have a drink of water before continuing the drive to his own home, she said.

Once inside her apartment, Ms. Brennan said, he pushed her onto a couch and forced himself on top of her. She said he pulled down her white V-neck T-shirt in order to put his mouth on her breasts before reaching behind her and shoving his hand down her pants and putting his fingers inside her vagina.

“I say, ‘Stop, why are you doing this?’ ” Ms. Brennan said. “And then I straight up said: ‘This is not consensual.’ ”

At that point, Ms. Brennan said Mr. Alvarez pulled off her pants and underwear and took off some of his clothes before thrusting onto her in a way that made her believe he was trying to force sexual intercourse. She said she was able to kick him off and run across the apartment to lock herself in the bathroom.

She said Mr. Alvarez left, and that she immediately called her husband, who at the time was in Sweden on a three-month Fulbright scholarship. She said she then called her best friend, who came to her apartment and stayed with her for several days.

Both her husband, Travis Miles, and friend, Katy Baldwin, confirmed Ms. Brennan told them she had been attacked. Her friend said Ms. Brennan was still trembling the evening after the assault. “She was exhibiting so many signs of shock,” Ms. Baldwin said. “She’s a very strong person. It was horrible to see her go through this, to continue to go through this.”

Ms. Brennan in Jersey City. When she found out prosecutors wouldn’t bring charges in her case, ‘I closed my office door and I just bawled,’ she said. ‘I felt like I wasn’t heard.’ Photo: Yana Paskova for The Wall Street Journal

The next day, Ms. Brennan called the police. The day after that, she went to the Jersey City Medical Center Emergency Department to be evaluated for sexual assault, hospital and police records show.

She FedEx-ed a short letter to Mr. Alvarez on April 17, which was reviewed by the Journal, saying he had sexually assaulted her and asking him to never contact her again. On the advice of a therapist, she wrote down a detailed account of the events.

Ms. Brennan spoke to Hudson County prosecutors and waited for months, calling in every so often to check up on the investigation.

Over the summer of 2017, Ms. Brennan signed up with the Murphy campaign as a volunteer policy adviser. She said she didn’t want to let the assault stand in the way of her goal of working for the campaign and Mr. Murphy’s administration, should the Democrat win. She said she wasn’t in the office every day but feared running into her alleged attacker, and did on occasion.

In late November, Ms. Brennan believed that the prosecutor, who she said had told her a decision was near, was close to charging Mr. Alvarez. By this point, Mr. Murphy had been elected New Jersey’s next governor and Ms. Brennan and Mr. Alvarez were both working on his transition committee.

Ms. Brennan believed Mr. Alvarez’s arrest would generate news coverage, and she allowed a friend that worked on the team to warn transition counsel that Mr. Alvarez might be charged.

A senior administration official said that members of the transition, including the governor’s chief of staff Pete Cammarano, were aware of a sexual-assault accusation against Mr. Alvarez. They didn’t know that the alleged victim was Ms. Brennan, the official said.

Transition officials learned that law enforcement had investigated the accusation against Mr. Alvarez and didn’t pursue charges, the governor’s spokesman said. “Following a clear background check, Mr. Alvarez received an offer of employment in state government,” he said.

Earlier this year, in March, Ms. Brennan told Mr. Murphy’s chief counsel, Matt Platkin, that she was sexually assaulted, allegedly by Mr. Alvarez. A senior administration official said Mr. Platkin referred the matter to the chief ethics officer in the governor’s office and recused himself from the investigation because he knew both Ms. Brennan and Mr. Alvarez. The ethics officer then referred the matter to the attorney general’s office, the official said. “We can confirm that the governor’s office conveyed information to the attorney general’s office regarding Mr. Alvarez,” a spokeswoman at the attorney general’s office said.

Mr. Platkin instructed Mr. Alvarez’s then boss, New Jersey Schools Development Authority Chief Executive Charlie McKenna, in June to tell Mr. Alvarez it would be a good idea for him to “separate himself” from state employment, the official said.

Mr. Alvarez told Mr. McKenna he would start looking for jobs, but a timeline for his departure wasn’t specified, Mr. McKenna said. “He wasn’t being fired, he wasn’t being ordered to leave,” he said. “It was just a conversation where I said, ‘I was told that this would be a good idea.’ ”

One of the senior administration officials said advising Mr. Alvarez to leave was legally the most they could do against an employee at a state authority.

No one told Ms. Brennan that Mr. Alvarez had been asked to leave, and on June 1 she emailed the governor and first lady. Jonathan Berkon, a partner at the law firm Perkins Coie LLP, who served as an attorney for the campaign, called and told her Mr. Alvarez would be leaving his job.

Three months later, Mr. Alvarez was still working at the development authority. Ms. Brennan said learning this prompted her to speak publicly about the alleged assault.

Ms. Brennan said she wants to see changes in how sexual-assault allegations are handled in New Jersey. She said it should be easier for prosecutors to pursue these cases in court, and the civil statute of limitations should be longer than two years. “People need to do better. Society needs to do better,” she said.

Corrections & Amplifications

Matt Platkin, the chief counsel for New Jersey Gov. Phil Murphy, instructed New Jersey Schools Development Authority Chief Executive Charlie McKenna to tell Al Alvarez to “separate himself” from government employment in June. A senior Murphy administration official previously told the Journal that the instruction came in April. (Dec. 21, 2018)

Write to Kate King at Kate.King@wsj.com