Interviews across the state found many Republican voters backing away from Ms. Cheney.

“I’m thinking, this time, Harriet Hageman,” said Marge Glick, 79, a retiree in Teton County, home of the Jackson Hole tourism area. “Liz Cheney, even though she says she’s following the Constitution, feels to me like she’s moving somewhat to the left.”

Ms. Cheney, the daughter of former Vice President Dick Cheney, with generations of ties to Wyoming, is now looking to court moderate Republicans living in the state’s urban areas such as Cheyenne, Laramie and Casper, as well as persuade Democrats and independents to change party affiliation to vote for her.

“I don’t typically vote in the Republican primary but I’m seriously considering changing my affiliation so I can vote for Liz…because she’s standing up for democracy,” said Ned Hutchinson, a 54-year-old apparel manufacturer who calls himself a Democrat by nature. He lives in Wilson, the mountain hamlet that Ms. Cheney also calls home.