A top state-level Republican questioned how Alicia Machado, the former Venezuelan beauty queen who says Donald Trump demeaned her, became a U.S. citizen this year, given a checkered past that included allegations of threatening a judge and involvement in a drive-by shooting.

Ms. Machado was not convicted in those incidents, but her credibility as an anti-Trump messenger has come under scrutiny after Democratic presidential nominee Hillary Clinton highlighted her story at Monday’s presidential debate, and the Clinton campaign hosted a press conference call for Ms. Machado to continue her attacks Tuesday.

“Her using Alicia Machado to try to malign Trump is another example of the system being broken,” Arizona Republican Party Chairman Robert Graham said. “Machado threatens to kill a judge, she’s a suspect in a drive-by shooting, she has known relations with a drug cartel boss Beltrán Leyval and she still becomes a U.S. citizen.”

The Associated Press reported in 1998 that Ms. Machado was seen speeding away in a car with her boyfriend after a murder outside a church in Venezuela. After the boyfriend was indicted for attempted murder, the judge in the case went on national television and said Ms. Machado threatened to kill him.

On CNN Tuesday anchor Anderson Cooper asked about the accusations, saying “Trump surrogates” were raising the issue. Ms. Machado brushed them aside.

“I have my past. Of course. Everybody has. Everybody has a past. And I’m not a saint girl,” Ms. Machado told CNN.

“That moment in Venezuela was wrong, was another speculation about my life because I’m a really famous person in my country, because I’m an actress there,” she said.

Democrats have seized on Ms. Machado as the latest example of Mr. Trump being rude to women and demeaning to Latinos — the GOP nominee allegedly also called Ms. Machado “Miss Housekeeping.”

Ms. Machado says Mr. Trump was mean to her after she won the 1996 title at the Miss Universe pageant owned by the billionaire businessman. She says he called her “Miss Piggy” and he made her attend a public gym workout session to demonstrate she was trying to get back into shape.

She says that rudeness harmed her self esteem. She became a U.S. citizen in August and said she was eager to vote against Mr. Trump.

Immigrant-rights group America’s Voice said Ms. Machado’s story “cut right to the heart of Trump and Trumpism” by exposing the GOP nominee as anathema to a pluralistic America.

“This is a moment of truth for who we are as a nation,” said Frank Sharry, executive director of the group.