(CNN) Donald Trump is in a spot all too familiar to other politicians, but unusual for him: He needs to regain his footing after a tough few days.

He swept into Washington Thursday to reassure House Speaker Paul Ryan and other Republican leaders that they can trust him to lead the party to victory in November.

But the narrative quickly shifted from Trump's savvy party outreach to his salacious past.

The next morning, The Washington Post's "John Miller" story dropped -- and suddenly the talk was all about whether Trump had once posed as his own spokesman.

The unconventional standard-bearer is running smack into conventional Washington vetting -- just as Trump's reluctant establishment allies try to quell a rebellion against him.

No third-party for elections

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Republican National Committee Chairman Reince Priebus spent Sunday swatting down efforts to launch a third-party presidential candidate to challenge Trump and Hillary Clinton, calling that strategy a " suicide mission.

The Washington Post had reported Saturday that 2012 Republican presidential nominee Mitt Romney -- who has emerged this cycle as a primary Trump antagonist -- made an unsuccessful attempt to recruit Nebraska Sen. Ben Sasse as a third-party presidential candidate to challenge Trump and Clinton.

A source with knowledge of the efforts confirmed them to CNN Sunday. Sasse, the freshman Republican, is one of Trump's most vocal critics on the right. Romney and Sasse have talked, the source said, and Romney encouraged Sasse to seriously consider a run.

Priebus was having none of that.

"They can try to hijack another party and get on the ballot, but, look, it's a suicide mission for our country because what it means is that you're throwing down not just eight years of the White House but potentially 100 years on the Supreme Court and wrecking this country for many generations," Priebus said on "Fox News Sunday," predicting that a conservative third-party candidate would split the Republican vote and ensure a Democrat wins the White House.

"So I think that's the legacy these folks will leave behind. I think it's very dangerous, and there's other ways to get assurances on the things that they're worried about," he said.

Trump 'to answer for' behavior around women

But even as Priebus urged the party to get behind Trump, he acknowledged that Trump "is going to have to answer for" his behavior around women.

The RNC chairman was asked on ABC's "This Week" about the Times report that Trump's private conduct with women has involved unwelcome advances and comments about women's figures.

"Look, I mean, these are things that he is going to have to answer for," Priebus said.

"But I also think there are things from many years ago and I think that, you know, as Christians, judging each other I think is -- is problematic. I think it's when people live in glass houses and throw stones is when people get in trouble."

The glass houses defense of Trump came one week after he attacked Clinton as an "enabler" of her husband's own behavior toward women.

But Priebus said he doesn't believe voters are judging Trump on his earlier personal life.

"I think people are judging Donald Trump as to whether or not he's someone that's going to go to Washington and shake things up. And that's why he's doing so well," he said.

Trump got a boost Monday from one of the women named in the Times story, who accused the newspaper of taking her words out of context.

"I'm very upset with the New York Times article because it was completely misleading," Rowanne Brewer Lane told CNN's John Berman and Kate Bolduan on "At This Hour." "They misled me, they took parts of what I said in at least a two-hour interview that they did exclusively with me and spun it and put a negative connotation on what I was saying."

Trump convention manager Paul Manafort also defended Trump by saying the voters don't care about these stories. This time, the topic was recordings from a People magazine interview with "John Miller."

"I couldn't tell who it is. If Donald Trump says it's not him, I believe it's not him," Manafort told CNN's Jake Tapper Sunday on "State of the Union."

Pressed on Trump's admissions in the past that he has used the names John Miller and John Barron, and that recordings of "John Miller" from an interview with People magazine sound like Trump, Manafort dismissed the question.

He said he's worked for Trump for six weeks and already uses language similar to his boss.

"I just know that he said it's not him," Manafort said. "I believe him. I don't even know the relevance of this, frankly."

Questions over tax returns

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Manafort similarly deflected Trump's refusal to release his tax returns -- a break with tradition and another controversy that has dogged the presumptive nominee in recent days. Manafort cited Trump's explanation that his returns since 2009 are still the subject of Internal Revenue Service audits.

"This is an issue the media is interested in," Manafort said. "It's not an issue the rest of America is interested in, frankly."

As The Washington Post's Chris Cillizza notes , the "John Miller" kerfuffle is exactly the sort of story that Trump's legions of loyal supporters have no interest in. In the panoply of Trump controversies that they have relished, rolled their eyes over and forgiven, posing as his own spokesperson in the distant past barely registers.

But it's also the sort of story that makes the Washington establishment extremely nervous. As Trump tries to reassure restive party regulars that he's a dependable standard-bearer, this oddity -- combined with reporting from the Times, CNN and other news outlets about his conduct toward women -- reinforces their anxieties.

Perhaps seeing an opening, President Barack Obama jumped into the fray Sunday to take aim at Trump, telling graduates during a commencement address at Rutgers University to tout their knowledge, not boast about their ignorance. He didn't name Trump -- but he didn't have to.

"In politics and in life, ignorance is not a virtue," he told the more than 12,000 graduates in New Brunswick, New Jersey. "It's not cool to not know what you're talking about. That's not keeping it real or telling it like it is. That's not challenging political correctness. That's just not knowing what you're talking about. And yet we've become confused about this."

Trump, never hesitant to vent his feelings on Twitter, trained his frustrations Sunday on a target that both outsiders and insiders revel in bashing: the media.

"The media is really on a witch-hunt against me," he tweeted. "False reporting, and plenty of it — but we will prevail!"

Eric Bradner and Kevin Bohn contributed to this report.