ANCHORAGE — Two of the most prominent women in Alaskan Republican politics — Senator Lisa Murkowski and Sarah Palin — appeared to be on divergent paths early Wednesday following the state’s special election and primary. Ms. Murkowski, 65, spurned by former President Donald J. Trump, advanced to the general election in November in the Senate race, according to The Associated Press. Ms. Palin, 58, who had Mr. Trump’s backing, also advanced in the fall for an open House seat but was trailing her Democratic opponent.Both races captured the fierce division among Republicans across the country and gave a glimpse into the independent and libertarian streak unique to Alaskan politics. They also underscored the surprising sway of Democrats in what has been a reliably red state, as well as the power of Native voters, a sizable electorate that does not predictably break for either party. The support of Native voters was key to the strong showings of both Ms. Murkowski and Ms. Palin’s main Democratic rival, Mary Peltola, a former state lawmaker who is Yup’ik and who would become the first Alaska Native in Congress if elected. More than 15 percent of Alaska’s population identifies as Indigenous.Still, final official results in the elections could take days and even weeks, as election officials in Alaska continue to collect and count mail-in ballots. The races Tuesday also tested a new complex voting system that allowed voters to rank their preferences in the special election. The process had rankled some Republicans who worried about losing power, but was seen by its proponents as encouraging candidates to appeal to voters beyond their base.In the Senate race, Ms. Murkowski has been in one of the toughest fights of her political career after voting to convict Mr. Trump in his impeachment trial following the Jan. 6, 2021, attack on the Capitol. Although she has lost support among Trump Republicans, she has attempted to forge a path to victory by solidifying a coalition of moderate Republicans, Democrats and independents that has helped keep her in office for three terms. She and Trump-endorsed Kelly Tshibaka advanced in a 19-way Senate primary. Ms. Murkowski was in the lead by three percentage points.Ms. Peltola, 48, took 37.8 percent of the vote in the special election to fill Alaska’s lone congressional seat through January, putting her more than five percentage points ahead of Ms. Palin, the state’s former governor and 2008 vice-presidential Republican nominee. Ms. Peltola was also leading Ms. Palin by nearly four percentage votes in the primary race to fill that seat beyond 2023. A win in the special election could provide a major boost in name recognition and momentum for Ms. Peltola, who has quickly risen to prominence since placing fourth in a June special election primary. On Tuesday, Ms. Peltola mingled with supporters at an Anchorage brewery as the results rolled in.“It’s just really overwhelming to see the kind of support that I’m getting,” she said. “I am hopeful.”More Coverage of the 2022 Midterm ElectionsLiz Cheney’s Lopsided Loss: The Republican congresswoman’s defeat in Wyoming exposed the degree to which former President Donald J. Trump still controls the party’s present — and its near future.2024 Hint: Hours after her loss, Ms. Cheney acknowledged that she was “thinking” about a White House bid, a prospect that would test the national viability of a conservative, anti-Trump platform.The ‘Impeachment 10’: With Ms. Cheney’s defeat, only two of the 10 House Republicans who voted to impeach Mr. Trump remain.Alaska Races: Senator Lisa Murkowski and Sarah Palin appeared to be on divergent paths following contests that offered a glimpse at the state’s independent streak.The House race began taking shape soon after the sudden death in March of Don Young, who represented Alaskans in Congress for nearly 50 years. As she has attempted to stage a political comeback, Ms. Palin has leaned on a solid base of support among evangelical conservatives and Trump devotees. She has shunned the establishment and mostly ignored the press. But a debate has brewed among Republicans over whether she is pursuing the seat in the name of public service or celebrity. Ahead of Tuesday, she and her top Republican challenger, Nick Begich III, had been trading barbs over their brands of conservatism and loyalties to Alaska. The infighting appeared to give Ms. Peltola an edge as she campaigned on bipartisanship and healing divisions.She and Ms. Palin have had a warm relationship since the two were expectant young mothers when Ms. Palin was governor and Ms. Peltola was still serving in the State Legislature. At a candidate forum hosted by The Anchorage Daily News, Ms. Palin even pointed to Ms. Peltola when asked who she would rank second on the ballot. On Tuesday, Ms. Peltola said Ms. Palin had texted her that morning to wish her well and remind her to dress warm.Sarah Palin at a rally in July hosted by former President Donald J. Trump in Anchorage. She is trying to make a comeback after more than a decade out of politics.Ash Adams for The New York TimesBut Ms. Palin seemed to mostly avoid everyone else. As national reporters flew into Anchorage and her hometown of Wasilla, her campaign did not respond to requests for interviews and did not release details about any election-night events. She posted a Facebook video of herself waving signs with volunteers in the early hours Tuesday. Later on a busy thoroughfare in central Anchorage, groups of Palin supporters and volunteers for other campaigns roamed in a final push to get voters to the polls.Decked out in Palin gear, Lisa Smith, 73, a retired educator, argued that Ms. Palin did not need the publicity. “The long-term Alaskans know her, and she has a history that is solid and caring,” she said.Mary Peltola, the only Democrat in the 22-candidate House primary, would become the first Alaska Native in Congress if elected. She was in the lead with about two-thirds of votes counted.Ash Adams for The New York TimesThe House race centered on abortion rights, the economy, climate change and the use of Alaska’s mineral resources. In the undertow was Mr. Trump, who made a rare visit to the state in July to promote Ms. Palin and Ms. Murkowski’s main challenger, Ms. Tshibaka. Ms. Palin appeared to retain a strong well of support in her hometown of Wasilla, a small city of 10,000 north of Anchorage, and in other parts of the state. Many of her most ardent admirers are conservative women who praise her accomplishments as a politician and as a mother, and see her as an answer to strong-minded and vocal women on the left, including House Speaker Nancy Pelosi and Representative Alexandria Ocasio-Cortez of New York.“People want to bring up that she resigned being governor, but there are reasons for that, and they were legitimate, and she was looking out for Alaskans,” said Melinda Michener, 62, an elementary-school teacher who has known Ms. Palin since Ms. Michener’s husband became a pastor at Ms. Palin’s childhood church.Yet, pollsters see a difficult climb for Ms. Palin given her dismal overall approval ratings. The Alaska Survey Research in late July found that 31 percent of registered Alaska voters viewed her positively and 61 percent viewed her negatively. In a different analysis, Republican pollster Matt Larkin believed it was most likely that Ms. Peltola or Mr. Begich would win the special election based on Ms. Palin’s low favorability numbers.Many voters disapproved of the persona and rhetoric that Ms. Palin adopted when she entered national politics in 2008 as the vice-presidential nominee for John McCain. Others argued that she had spent most of her time since then in the lower 48 states, a particularly stinging affront to many Alaskans who often pride themselves as being separate from the rest of the United States. Nick Begich III, the Republican scion of the state’s most prominent Democratic political family, at a campaign event in Wasilla, Alaska.Ash Adams for The New York TimesMr. Begich, 44, the founder and chief executive of a software development company, also faced accusations of being an outsider. He was born in Alaska but grew up in Florida after his parents split. He sought to define himself as a young and idealistic fiscal conservative, despite sharing a last name with the best-known Democratic family in the state. His true ideological opponent over the direction of the state was Ms. Peltola, who has strongly championed abortion rights, called for higher taxes on the wealthy and has sought an approach to development of Alaska’s resources focused on sustaining communities over corporate interests. As a Yup’ik woman, she has said a “pro-family ethic” shapes her identity.In the Senate race, Ms. Tshibaka has sought to capitalize on longtime conservative frustrations with Ms. Murkowski, including her vote in 2017 against repealing the Affordable Care Act and her support of Deb Haaland for Interior Secretary under the Biden administration.Kelly Tshibaka campaigning on Tuesday in Alaska. She sought to run to the right of Ms. Murkowski in their Senate race.Ash Adams for The New York TimesOn Tuesday in Anchorage, hours before the polls closed, she and Ms. Murkowski waved signs and cheered at honking cars on opposite sides of the street. “It is a choice between the senator Joe Biden wants to have and the senator for Alaska values and Alaska’s interests,” Ms. Tshibaka said, as supporters behind her screamed, “Vote for Kelly.”Ms. Murkowski has maintained that there is still a place for her bipartisan relationships and her independent streak. The open primary system, coupled with a general election in November that will allow voters to once more rank their choices, is widely seen as designed to benefit more centrist candidates like her.This is not the first time Ms. Murkowski has found herself in a fight for political survival. In 2010, after she was defeated in the Republican primary, she beat a Tea Party candidate in a long-shot run for re-election as a write-in candidate. Her campaign team at the time emblazoned her name on silicon wristbands to help voters remember how to properly spell her name on the ballot. After her victory, she had a replica made in gold. “I’ve worn it on my wrist every day since 2010 to remind me that I was not returned to the United States Senate in a traditional way,” she told reporters Friday after meeting with voters in Talkeetna. “I returned at the request of Alaskans.”Emily Cochrane More