CHEYENNE, Wyoming — Congresswoman Elizabeth Cheney, a leader in the Republican resistance to President Trump, is fighting to save her seat in the House on Tuesday as voters weigh in on the direction of the GOP.

Ms. Cheney’s team is bracing for a loss against a Trump-backed challenger in the state in which he won by the largest of margins during the 2020 campaign.

Win or lose in deep-red Wyoming, the 56-year-old daughter of a vice president is vowing not to disappear from national politics as she contemplates a 2024 presidential bid. 

Yet in the short term, Cheney is facing a dire threat from her Republican opponent, a Cheyenne ranching industry attorney, Harriet Hageman, who has harnessed the full fury of the Trump movement in her bid to expel Ms. Cheney from the House.

“I’m still hopeful that the polling numbers are wrong,” said a Wyoming state representative and vocal Cheney ally, Landon Brown. “It’ll be a crying shame really if she does lose. It shows just how much of a stranglehold that Donald Trump has on the Republican Party.”

Tuesday’s contests in Wyoming and Alaska offer one of the final tests for Trump and his brand of hard-line politics ahead of the November general election. So far, the former president has largely dominated the fight to shape the GOP in his image, having helped install loyalists in key general election matchups from Arizona to Georgia to Pennsylvania.

This week’s contests come just eight days after the FBI executed a search warrant at Mr. Trump’s Florida estate, recovering 11 sets of classified records. Some were marked “sensitive compartmented information,” a special category meant to protect the nation’s most important secrets. The Republican Party initially rallied behind the former president, although the reaction turned somewhat mixed as more details emerged.

In Alaska, a recent change to state election law gives a periodic Trump critic, Senator Murkowski, an opportunity to survive the former president’s wrath, even after she voted to convict him in his second impeachment trial.

The top four primary Senate candidates in Alaska, regardless of party, will advance to the November general election, where voters will rank them in order of preference.

In all, seven Republican senators and 10 Republican House members joined every Democrat in supporting Mr. Trump’s impeachment in the days after his supporters stormed the Capitol as Congress tried to certify President Biden’s victory.

Just two of those 10 House members have won their GOP primaries this year. The rest have lost or declined to seek reelection. Ms. Cheney would be just the third to return to Congress if she defies expectations on Tuesday.

And Ms. Murkowski is the only pro-impeachment senator running for reelection this year.

She is facing 18 opponents — the most prominent of which is Republican Kelly Tshibaka, who has been endorsed by Mr. Trump — in her push to preserve a seat she has held for nearly 20 years. Mr. Trump railed against Ms. Murkowski on social media and in her home state of Alaska, where he hosted a rally with Ms. Tshibaka last month in Anchorage.

In contrast to vulnerable Republican candidates who cozied up to Mr. Trump in other states this summer, Ms. Murkowski continues to promote her bipartisan credentials.

“When you get the ideas from both sides coming together, little bit of compromise in the middle, this is what lasts beyond administrations, beyond changes in leadership,” the Republican senator said in a video posted on social media over the weekend. “This is what allows for stability and certainty. And it comes through bipartisanship.”

On the other side of the GOP’s tent, the former vice-presidential nominee, Governor Palin hopes to spark a political comeback on Tuesday.

Endorsed by Mr. Trump, she finished first among 48 candidates to qualify for a special election seeking to replace Congressman Don Young, who died in March at age 88, after 49 years as Alaska’s lone House member. 

Ms. Palin is actually on Tuesday’s ballot twice: once in a special election to complete Young’s term and another for a full two-year House term starting in January. She’s running against Republican Nick Begich and Democrat Mary Peltola in the special election and a larger field in the primary.

Ever an outsider, Ms. Palin spent recent days attacking Ms. Murkowski, a fellow Republican, and those who instituted the open primary and ranked-choice voting system in 2020.

“I’ve said all along that ranked-choice voting was designed to benefit Democrats and RINOs, specifically Sen. Lisa Murkowski (who stood no chance of winning a Republican nomination) along with other political dynasty family members in Alaska,” Ms. Palin wrote in a recent statement calling for the law’s repeal.

Back in Wyoming, Ms. Cheney’s political survival may depend upon persuading enough Democrats to cast ballots in her Republican primary election. While some Democrats have rallied behind her, it’s unclear whether there are enough in the state to make a difference. Mr. Biden earned just 26 percent of Wyoming’s vote in 2020.

Many Republicans in the state — and in the country — have essentially excommunicated Ms. Cheney because of her outspoken criticism of Mr. Trump. The House GOP ousted her as the No. 3 House leader last year. And more recently, the Wyoming GOP and Republican National Committee censured her.

Ms. Cheney has seemingly welcomed defeat by devoting almost every resource at her disposal to ending Mr. Trump’s political career since the January 6 riot.

She emerged as a leader in the congressional committee investigating Mr. Trump’s role in the January 6 attack, giving the Democrat-led panel genuine bipartisan credibility. She has also devoted the vast majority of her time to the committee instead of the campaign trail back home, a decision that still fuels murmurs of disapproval among some Wyoming allies. And she has closed out the primary campaign with an unflinching anti-Trump message.

“In our nation’s 246-year history, there has never been an individual who was a greater threat to our republic than Donald Trump,” Vice President Cheney said in a recent ad produced by his daughter’s campaign.

He continued, “There is nothing more important she will ever do than lead the effort to make sure Donald Trump is never again near the Oval Office.”