The unfurling scandal has put Mr. Hawley in an unwelcome bind: He is both investigating the governor for unrelated campaign violations and being attacked by a barrage of Democratic television ads trying to link him with the governor in a cloud of impropriety.

While Missouri has become a solidly Republican state — Mr. Trump won here by 19 percentage points — Republicans are facing the potential of both a criminal trial and a messy impeachment fight involving their standard-bearer. Mr. Greitens, who has denied the charges, appears determined to fight on, raising fears among party leaders that a prolonged battle could depress Republican turnout and neutralize some of Mr. Hawley’s outsider appeal.

And as a backlash against Mr. Trump and male misbehavior is expected to push record numbers of women to the polls, Ms. McCaskill is practiced in the art of exploiting such moments.

“Right now, Greitens is a slow-moving disaster, and one of the people who can pay a price — is it 4 percent? Two percent? I don’t know — is Josh Hawley,” said Ed Martin, a former chairman of the state Republican Party.

It was not supposed to be this way. Having watched furiously in 2012 as Ms. McCaskill — thought to be all but certain to lose — waded directly into a crowded Republican primary race to help lift Mr. Akin to the nomination, Republicans were intent on giving the senator no such openings.

Ms. McCaskill’s campaign weighed in during that race with an advertisement laying out Mr. Akin’s conservative credentials, suggesting that Missouri’s “true conservative” was just “too conservative.”