NEW YORK (AP) — Leading Republicans are entering the final month of the midterm campaign increasingly optimistic that a Senate majority is within reach even as a dramatic family fight in Georgia clouds one of the party's biggest pickup opportunities.

And as some Democrats crow on social media about apparent Republican setbacks, party strategists privately concede that their own shortcomings may not be outweighed by the GOP's mounting challenges.

The evolving outlook is tied to a blunt reality: Democrats have virtually no margin for error as they confront the weight of history, widespread economic concerns and President Joe Biden's weak standing. There is broad agreement among both parties that the Democrats' summertime momentum across states like Nevada, Pennsylvania and Wisconsin has eroded just five weeks before Election Day.

"There's reason to be apprehensive, not reason to be gloomy," veteran Democratic strategist James Carville said. "It looked like at the end of August we had a little momentum. I don't know if we've regressed any, but we're not progressing in many places."

That tepid outlook comes even as Republicans confront a series of self-imposed setbacks in the states that matter most in the 2022 midterms, which will decide the balance of power in Congress and statehouses across the nation.

None has been more glaring than Herschel Walker's struggles in Georgia, where the Republican Senate candidate's own son accused him of lying about his personal challenges — including a report from The Daily Beast alleging that the anti-abortion Walker paid for a girlfriend's abortion in 2009. Walker called the accusation a "flat-out lie" and said he would sue.

Walker had not taken legal action as of late Tuesday, but he repeated his denials Wednesday morning during a Fox News interview, even as he talked generally of a difficult past as a husband and father. Shown an image of the "get well" card the Daily Beast reported that he sent to the girlfriend — which was signed with an "H," not his full signature — Walker said, he doesn't sign cards with just an initial.

The Republican establishment, including the Sen. Mitch McConnell-aligned Senate Leadership Fund, and former President Donald Trump himself remained staunchly behind Walker on Tuesday in his bid to oust first-term Democratic Sen. Raphael Warnock. A Walker campaign adviser said the candidate has raised at least $500,000 since he first responded publicly to The Daily Beast report.

"If you're in a fight, people will come to your aid," said Steven Law, head of the Senate Leadership Fund and a close ally of McConnell, R-Ky.

Law said the Georgia race had grown increasingly competitive despite the Democrats' focus on Walker's personal life. And looking beyond Georgia, Law said the political climate was predictably shifting against the party that controls the White House, as is typically the case in midterm elections.

"It certainly seems that voters are returning to a more traditional midterm frame of mind," Law said.

Should Republicans gain even one Senate seat in November, they would take control of Congress' upper chamber — and with it, the power to control judicial nominations and policy debates for the last two years of Biden's term. Leaders in both parties believe Republicans are likely to take over the House.

Even facing such odds, it's far too soon to predict a Republican-controlled Congress.

Democrats remain decidedly on offense and are spending heavily to try to flip Republican-held seats in Pennsylvania, Wisconsin and North Carolina. Voter opposition to the Supreme Court's decision this summer to strip women of their constitutional right to an abortion has energized the Democratic base and led to a surge in female voter registrations.

Republicans are most focused on Democratic incumbents in Arizona, Georgia, New Hampshire and Nevada, although Republican officials believe that underwhelming Trump-backed nominees in Arizona and New Hampshire have dampened the party's pickup opportunities.

"The Republican candidates they're running are too extreme," said J.B. Poersch, who leads the pro-Democrat Senate Majority PAC. "I think this is still advantage Democrats."

Meanwhile, conditions in the top battleground states are rapidly evolving.

In Pennsylvania, Republican Senate nominee Mehmet Oz faced difficult new questions this week raised by a Washington Post article about the medical products he endorsed as a daytime television star. Another news report by the news site Jezebel detailing how his research caused hundreds of dogs to be killed rippled across social media.

Still, Democratic officials acknowledge the race tightened considerably as the calendar shifted to October. And while there is disagreement within the White House, some officials there are concerned about Democratic nominee John Fetterman's stamina as he recovers from a May stroke.

"Senate Republicans had a very bad start to October, but we know each of our races will be tight and we're going to keep taking nothing for granted," said Michigan Sen. Gary Peters, who leads the Senate Democrats' campaign arm.

The GOP Senate candidates' latest challenges in Georgia and Pennsylvania dominated social media Monday and Tuesday, according to data compiled by GQR, a public opinion research firm that works with Democratic organizations.

News stories about Walker's abortion accuser and Oz's animal research had the first- and second-highest reach of any news stories on Facebook and Twitter since they surfaced Monday, topping content related to the television show "Sons of Anarchy," another report about Planned Parenthood mobile abortion clinics and news about Kanye West. GQR used the social listening tool NewsWhip, which tracks over 500,000 websites in more than 100 languages roughly in real time.

In swing-state Nevada, the rhetoric from Democratic Sen. Catherine Cortez Masto has become increasingly urgent in recent days as she fends off a fierce challenge from former state Attorney General Adam Laxalt. Within the White House, there is real fear that she could lose her reelection bid, giving Republicans the only seat they may need to claim the Senate majority.

"We have a big problem, friend," Cortez Masto wrote in a fundraising appeal Tuesday. "Experts say that our race in Nevada could decide Senate control — and right now, polling shows me 1 point behind my Trump-endorsed opponent."

Democrats and their allies continue to hope that backlash against the Supreme Court's abortion decision will help them overcome historical trends in which the party controlling the White House almost always loses seats in Congress. Democrats, who control Washington, are also facing deep voter pessimism about the direction of the country and Biden's relatively weak approval ratings.

The traditional rules of politics have often been broken in the Trump era. In past years, Republicans may have abandoned Walker. But on Tuesday, they linked arms behind him.

Law, of the Senate Leadership Fund, said he takes Walker at his word that he did not pay for a former girlfriend's abortion, despite apparent evidence of a "Get Well" card with Walker's signature and a check receipt.

He said voters believe that "Walker may have made mistakes in his personal life that affected him and his family, but Warnock has made mistakes in public life in Washington that affected them and their families."

There were some signs of Republican concern on the ground in Georgia, however.

Martha Zoller, a popular Republican radio host in north Georgia and one-time congressional candidate, told her audience Tuesday that the latest allegations require Walker to reset his campaign with a straightforward admission about his "personal demons" and what he's done to overcome them.

"He needs to fall on the sword. 'I was a dog. ... And I have asked forgiveness for it,'" she said, detailing the kind of message she believes Walker must give voters. "It would be so refreshing to have somebody just tell the truth."

Walker attempted his version of that strategy Wednesday on Fox News.

"It's like they're trying to bring up my past to hurt me," he said, before quoting Christian New Testament text. "I'm a sinner. We all sin before the glory of God."

Yet Walker insisted his past transgressions don't include encouraging and paying for an abortion. "Everyone is anonymous, and everyone is leaking, and they want you to confess to something you have no clue about," he said.

Veteran Democratic strategist Josh Schwerin warned his party against writing off the Georgia Republican.

"I wouldn't say Walker is done. Over the last couple of cycles we've certainly seen Republican candidates survive things that are not supposed to be survivable," Schwerin said. "There are a lot of close races, and the dynamics of this election are difficult to predict. Everybody is expecting multiple shifts in momentum between now and Election Day."

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Associated Press writers Zeke Miller in Washington and Bill Barrow in Atlanta contributed to this report.