Abortion has been a losing issue at the polls for Republicans across the country since the Supreme Court’s decision to overturn Roe v. Wade. But now in Virginia, which holds elections in early November, the party thinks it has hit upon a formula to stop the electoral drubbings.

Legislative races across the state will offer a decisive test of a strategy led by Gov. Glenn Youngkin, who has united Republicans behind a high-profile campaign in support of a ban on abortion after 15 weeks with exceptions for rape, incest and the life of the mother. The party calls it a “common sense” position, in contrast to Democrats, who it says “support no limits.”

The strategy is meant to defuse Republicans’ image as abortion extremists, which led to losses in last year’s midterms and threatens further defeats next month in an Ohio referendum and the Kentucky governor’s race.

The approach is similar to one being pursued by Republican Senate candidates in battleground states like Arizona, Pennsylvania and Michigan, where the party has been open to some exceptions, a stance that research shows is more popular than an outright ban.

Virginia Republicans aren’t looking to win over abortion-rights supporters so much as they want to neutralize the party’s disadvantage with swing voters. The hope is that these voters will prioritize a competing set of issues such as crime and the economy, on which Republicans have an advantage in some polls.

All 140 seats in the state’s General Assembly are on the ballot this fall, with Republicans looking to take full control. Democrats have made the threat to abortion rights their No. 1 issue, pouring money into ads and looking to motivate voters in an off-year election with President Biden’s unpopularity dimming enthusiasm.

If Republicans take majorities in both legislative chambers under Mr. Youngkin, a governor with national ambitions, it would clear the way for Virginia to become the last Southern state to sharply restrict abortions.

Since mid-October, Mr. Youngkin’s political action committee has run a $1.4 million ad campaign taking the offensive on the issue. Accusing Democrats of “disinformation,” it promotes the 15-week limit with exceptions as “reasonable” and “common sense.”

The Youngkin ad, targeted at swing districts and echoed by the ads of individual Republicans running, shatters the formula of most G.O.P. candidates in battleground states after the reversal of Roe v. Wade in 2022, who dodged abortion in midterm races and often lost.

“We’re just simply not going to repeat 2022,” said Zack Roday, the coordinated campaigns director for Mr. Youngkin’s political group.

Kaitlin Makuski, the political director of Susan B. Anthony Pro-Life America, a national anti-abortion group with close ties to Mr. Youngkin, said that if Virginia Republicans prevailed this year, it would be a clear signal to candidates in 2024 that leaning into a 15-week ban can be successful.

“He and his team looked back at what they saw in 2022 and realized we can’t continue burying our head in the sand,” she said of the governor. “We need to move forward. This is a great template to follow.”

Existing Virginia laws, which Democrats want to keep in place, allow abortions with no restrictions through the second trimester, about 26 weeks, and thereafter if three doctors certify that a pregnancy would “irremediably impair” the mother’s health.

“Virginia has in place a law that parallels Roe v. Wade, that allows women to have freedom of choice to make their own health decisions,” said Senator Mamie Locke, chairwoman of the Virginia Senate Democratic caucus. “Why do you have to change the law to this 15-week ban? What’s ‘reasonable’ about that?”

Democrats point to other Republican-led states that have banned abortion in almost all circumstances and say a 15-week limit is a ruse that will give way to stricter limits if Republicans gain full control of government. Last year, Mr. Youngkin told conservative activists that he would “happily and gleefully” sign any bill to “protect life.” The governor has insisted he is only interested in a 15-week limit.

A 15-week ban, just past the first trimester of pregnancy, polls well in some surveys. A Gallup poll this year found that 69 percent of U.S. adults support abortion in the first trimester, but support falls to just 37 percent in the second trimester.

In a Washington Post-Schar School poll this month, Virginia voters were equally divided on the 15-week ban with exceptions: 46 percent supported such limits and 47 percent opposed them.

But in an illustration of how abortion polling can yield conflicting results, 51 percent of voters in the poll said they trusted Democrats to do a better job handling abortion vs. 34 percent who trust Republicans.

Even if a 15-week ban doesn’t convert many voters for whom abortion rights are a top issue — and most of those who say so are Democrats — the G.O.P. bet is that they can neutralize the issue with independent voters. In the Washington Post poll, independents said they trusted Democrats more on abortion, but Republicans more than Democrats on crime and the economy.

“Youngkin thinks the Republicans have an advantage on a set of issues people care about. They don’t on abortion, so they have to reduce the level of threat so people don’t vote on that issue,” said Bob Holsworth, the founding director of the School of Government at Virginia Commonwealth University. “He wants them to vote on these other issues where he thinks he’s in better shape.”

Danny Diggs, a Republican running for State Senate in a crucial district around Newport News, enlisted his adult daughter Michelle to record an ad about his support for a 15-week limit. “Take it from me,” she says in the ad, her father “will not cater to the extremes.”

Over the weekend, as Mr. Diggs, a retired sheriff, greeted voters at a seafood festival in Poquoson, a town on Chesapeake Bay, he said he would vote against any bill limiting abortion earlier than 15 weeks. “I’m good with the 15 weeks, that’s what I’ve told people,” he said.

Charles Salas, 53, who is retired from the Army, greeted Mr. Diggs as he stood beside a Republican Party tent and liked what the candidate had to say. On abortion, he sounded more conservative than Mr. Youngkin’s proposed 15-week cutoff. “I haven’t decided how early but I think it should be early enough,” he said. “I don’t believe it should be on demand and I shouldn’t have to pay for it,” he said.

Ann Holland, a 58-year-old school district employee, said she was undecided in the election, but on the abortion issue, she wanted women to have broad leeway to make a choice. “I was in my third month and didn’t know,” she said with a laugh. “No morning sickness, no nothing.”

Mr. Diggs said that in knocking on the doors of thousands of Republicans and independent voters, the top issues he heard about were public safety and education. Abortion did not often come up. “I don’t think it’s as important as the Democrats hope that it is,” he said.