Worried about inflation and dissatisfied with President Biden, many moderate women have been drifting away from Democrats. Now the party hopes the fight for abortion rights will drive them back.GRAND RAPIDS, Mich. — As Gov. Gretchen Whitmer prepared to kick off a round-table discussion about abortion rights at a brewery recently, Alisha Meneely sat at one corner of the table, feeling politically abandoned.Ms. Meneely voted for Donald Trump in 2016 before supporting President Biden in 2020, she said. Now, she is struggling with both parties, gravely disappointed in Mr. Biden’s leadership but anguished by what she sees as a Republican lurch toward extremism, with little room for disagreement — especially on abortion rights.“This scares me a lot,” said Ms. Meneely, 43, who described herself as a “pro-choice Republican” in an interview shortly before the Supreme Court overturned Roe v. Wade.A few days later, as many Republican officials embraced the far-reaching implications of the decision, she was unequivocal. “This,” Ms. Meneely said, “is not my party.”After struggling for months against daunting political challenges, Democrats have a new opening to engage moderate women like Ms. Meneely, who have been critical to the party’s recent victories but are often seen as swing voters this year, according to interviews with more than two dozen voters, elected officials and party strategists across the country.From the suburbs of Philadelphia and Grand Rapids to more conservative territory in Nebraska, there are early signs that some voters who disapprove of Mr. Biden also increasingly believe that Republicans have gone too far to the right on a range of issues, particularly abortion.Democrats see a new opportunity to engage dissatisfied voters in the fight over abortion rights. Jamie Kelter Davis for The New York TimesIt’s a dynamic with the potential to shape statewide races and some House contests, and one that crystallizes a central tension of the midterm elections as Democrats test whether efforts to define today’s Republicans as extremist can mitigate the political headwinds they confront.High inflation remains the overriding concern for many voters, and Republicans are betting that most Americans will vent about pocketbook frustrations above all else. Mr. Biden has long struggled with anemic approval ratings. Americans also overwhelmingly believe the country is headed in the wrong direction, another troubling sign for the party in power. And some Democrats doubt that even something as significant as the overturning of Roe will dramatically alter the political environment.For many Americans, economic struggles outweigh abortion rights as the top issue.Sarah Silbiger for The New York Times“Does it have an effect? Absolutely,” said Chuck Rocha, a Democratic strategist. “Does it fundamentally change the landscape? No. Not in an off-year election, when your president’s approval rating is below 40 percent and gas is $5 a gallon.”Those crosscurrents all converged last week at a few shopping centers in Warrington, Pa., in Bucks County outside Philadelphia. It’s a swing township within a swing county in the nation’s ultimate swing state. The next governor and a Republican-controlled legislature will most likely determine access to abortion, after the Supreme Court’s recent decision handed control over abortion rights back to the states.Sophia Carroll, 22, said that growing up, some of her friends were engaged in anti-abortion activism. Citing her Catholic upbringing, Ms. Carroll, a registered Republican, said she felt mixed emotions when Roe was overturned. But she intended to vote for Democrats this fall, “just because of this issue” of protecting abortion rights.From Opinion: The End of Roe v. WadeCommentary by Times Opinion writers and columnists on the Supreme Court’s decision to end the constitutional right to abortion.David N. Hackney, maternal-fetal medicine specialist: The end of Roe “is a tragedy for our patients, many of whom will suffer and some of whom could very well die.”Mara Gay: “Sex is fun. For the puritanical tyrants seeking to control our bodies, that’s a problem.”Elizabeth Spiers: “The notion that rich women will be fine, regardless of what the law says, is probably comforting to some. But it is simply not true.”Katherine Stewart, writer: “Breaking American democracy isn’t an unintended side effect of Christian nationalism. It is the point of the project.”“As someone who knows other women who have had to make the decision to choose, it’s a very personal and very intimate decision,” she said in an interview at an outdoor shopping center.Ms. Carroll pointed out Justice Clarence Thomas’s concurring opinion, which suggested that the court should revisit its cases establishing rights to same-sex marriage, same-sex consensual relations and contraception.“Are they going to ban birth control next?” she said.There is limited polling that captures attitudes after the Supreme Court decision, and none of it predicts how voters will feel in November. A recent survey from NPR, PBS NewsHour and Marist found that 56 percent of adults surveyed opposed the decision and 40 percent supported it. Among people in suburbs, which in recent years have been home to many moderates and swing voters, 57 percent said they mostly support abortion rights; only a third said they mostly oppose abortion rights. Among women in the suburbs and small cities, support for abortion rights jumped to 61 percent.Another survey from Morning Consult and Politico found that among suburban voters, around 60 percent said it was very or somewhat important to support a candidate in the midterm elections who backs abortion access; roughly 40 percent said it was very or somewhat important to support a candidate who opposes that access.But polls have also consistently shown that the economy and inflation remain top issues for many Americans. And many voters are inclined to take their frustration about cost-of-living concerns out on the Democrats.“The economy is always going to be the biggest thing for me,” Diane Jacobs, 57, said in an interview outside a Wegmans grocery store in Warrington. Ms. Jacobs, who said that she typically votes for Republicans, identifies as “pro-life” but does not believe abortion should be illegal. She also voted for Mr. Biden in 2020, she said, as an antidote to divisiveness. But Ms. Jacobs said she would not do so again and planned on supporting Republicans this year.“Just look at inflation,” she said.Some voters are not yet aware of the implications of overturning Roe, which are unfolding day-by-day and state-by-state. Democrats may have room to expand their support on the issue as voters learn more. Republicans, however, may ultimately benefit if many voters who disagree with the decision don’t dive in on the details. Ms. Jacobs said she had not heard of Republicans in the area who wanted to outlaw the procedure.“If there was a presidential candidate who said they wanted to outlaw it in every single case, I don’t know that I’d vote for that person,” she said. “That’s pretty extreme.”Josh Shapiro, the Democratic nominee for governor of Pennsylvania, has promised to veto “any bill that would restrict abortion rights.”Hannah Beier/ReutersAbortion is now banned in at least eight states, with few exceptions allowed. Some legal challenges are underway, and more bans are expected to take effect soon. Doug Mastriano, the far-right Republican nominee for governor in Pennsylvania, has sponsored a roughly six-week abortion ban and has indicated interest in further restrictions, saying life begins at conception. Asked whether he believes in exceptions for rape, incest or the life of the mother, he replied at a debate, “I don’t give a way for exceptions.”Josh Shapiro, the Democratic nominee, has promised to veto “any bill that would restrict abortion rights.”The Pennsylvania governor’s race is one of several, including governor’s contests in Michigan and Wisconsin, that could directly affect abortion rights in battleground states.Barrie Holstein, 58, said she felt a new sense of political urgency. Ms. Holstein, who lives in Dresher, Pa., declined to say how she voted in 2020. She said she does not always vote in midterm elections and was often open to candidates of both parties. But this year, she said, she intended to vote for candidates who backed abortion rights and gun control.“I’m not political,” she said. “But it’s enough. I’m pissed. I’m pissed about gun control and I’m pissed about abortion. I really am.”Strategists in both parties are still trying to quantify how many voters like Ms. Holstein are out there.In a small private focus group of suburban swing voters last week sponsored by progressive organizations, a clear majority of participants said the Roe decision would hold either a lot or a medium amount of weight when considering how to vote in upcoming elections.But in one warning sign for Democrats, at least one participant said she felt it was “too late” — the party in power had already failed to protect abortion rights, so she would be weighing a broader set of issues.While some Republicans see openings to paint Democrats as radical on the issue of abortion rights late into pregnancy, many officials have largely sought to keep their focus on cost-of-living matters and on Mr. Biden.“I would be surprised if an energized Democratic electorate overcame the dead-weight anchor of a 40 percent job approval for a Democratic president,” said Whit Ayres, a Republican strategist. “But it might make some races closer than they would otherwise have been.”That may have been the case in a recent Nebraska special election, when a Democratic candidate did better than expected in a heavily Republican-leaning district. Turnout was just under 30 percent of registered voters.Jane Kleeb, Nebraska Democratic Party chairwoman, conducting party business remotely at the Lancaster County Democrats headquarters in Lincoln, Neb., in 2020.Walker Pickering for The New York Times“This is real and resonating and you feel it on the ground,” said Jane Kleeb, the chairwoman of the Nebraska Democratic Party. “Folks, I think, in the Midwest, really respect people’s privacy. Ranchers always say, ‘If it doesn’t bother the cattle, it doesn’t bother me.’ That mentality is very much alive, I think, in voters’ minds.”Last week, Ms. Meneely of Michigan — who has a background in government work and engages in efforts to combat human trafficking and online exploitation of children — said that she had decided to vote for Ms. Whitmer, the Democratic governor.She also said she would support Representative Peter Meijer, a Republican who applauded the Roe decision, in his primary. Ms. Meneely noted his willingness to challenge Mr. Trump. (He was one of 10 House Republicans to vote for impeachment after the Capitol riot.)But she sounded open to persuasion in general election contests.“Right now,” she said, “I am so ticked at the Republican Party.” More