LOGAN, Utah — On a night when Donald J. Trump’s campaign was staggering under new accusations that he had groped and sexually assaulted women, 900 voters streamed into a middle school auditorium in this reliably Republican and mostly Mormon city, desperate to hear something different.

“This is the only time in my life I’ve not known what to do,” said Holly Shapiro, 54, a conservative who has never missed a presidential election, but may stay home in November. Her daughter Nicole, sitting next to her, described Mr. Trump as profane and immoral, and Ms. Shapiro nodded, saying, “I want to vote for a moral, upstanding person.”

Backstage, Evan McMullin was getting ready to make his pitch that he was just that candidate.

He has little money and no major-party backing, and he is little known outside Utah, his home state. But lately, Mr. McMullin, 40, has been drawing big crowds in Utah and thousands of online supporters who see his moonshot run for president as a final refuge for “Never Trump” conservatives. Some polls show him turning the battle here into a three-way race with Mr. Trump and Hillary Clinton.

“People come up and they just thank us for running, in a relieved sort of way,” Mr. McMullin said at his campaign offices on a tree-lined street in downtown Salt Lake City. “Some people say that they were praying for another option.”