“There’s a real likelihood that they not only win the House, but they win it by 10 or 12 more seats than they need,” said Senator Lindsey Graham of South Carolina, voicing publicly what many Republican officials have begun to acknowledge privately this summer.

Mr. Graham said a Democratic takeover was no sure thing, noting that “in the era of Trump, things can change in 24 hours, for good or bad.” But he said any Republican in a remotely competitive district could face a difficult general election. “We’re bleeding among women and the enthusiasm factor for Democrats is worth 7 or 8 points, and sometimes more,” he said, using political jargon to describe just how deep into Republican territory the battlefield might stretch.

“If I was a House guy in an R+10 or less seat I’d be getting on the phone and raising money and putting a sign on my dog.”

[Here are key takeaways from Tuesday’s voting.]

In Kansas, Republicans faced uncertainty Wednesday not only about their candidate for governor this fall, but also whether they were creating a political opportunity for Democrats to win the office in November. Should the party ultimately nominate Kris W. Kobach, its hard-right candidate for governor, it could also undermine Republicans in congressional races down ballot.

With all Kansas precincts reporting Wednesday morning, Mr. Kobach, the secretary of state, was ahead of Gov. Jeff Colyer by just 191 votes out of more than 311,000 Republican ballots cast. The results were likely to remain in flux for at least several days.

And in another race where Republicans could be vulnerable in November, Sharice Davids, a lawyer who is Native American, won the Democratic congressional primary on Wednesday in a swing district that surrounds Kansas City. She will face Representative Kevin Yoder in November. It is one of two Republican-held districts in the state, along with a more rural open seat next door, where Democrats are threatening to shave down the House majority.