And if vulnerable lawmakers turn inward, that could make passing controversial legislation, like an overhaul of the health care or tax system, even more difficult.

Such a vicious cycle, retrenchment ensuring inaction, could only further demoralize grass-roots Republicans, deteriorating the party’s standing even more. That malaise has already hobbled Republicans in Georgia, forcing national “super PACs” to spend heavily to aid Ms. Handel while Mr. Ossoff has raised over $24 million on his own, mainly with support from small donors.

“We actually have to have victories,” Mr. Rooney said.

Notably, Mr. Ossoff has made opposition to the American Health Care Act, the health care bill approved last month by House Republicans, a signature issue of his campaign. And while Mr. Ossoff has run on a centrist message over all, Democrats have run advertisements targeting liberal-leaning voters, especially African-Americans, with appeals to send a message to Mr. Trump: Republicans on Saturday circulated a picture of a truck parked in the district with a sign reading, “Hold Donald Trump Accountable, Vote Tuesday, June 20.”

“First and foremost, this is a referendum on the Trump presidency,” said Representative Tom Cole, Republican of Oklahoma. “The stakes here are not just for the House, the stakes are for the Trump presidency.”

In a briefing last week to House Republicans, Representative Steve Stivers of Ohio, the head of the House campaign arm, did not guarantee an election victory. “He said, ‘It’s going to be a late night,’” Mr. Cole recalled.

On the Democratic side, elected officials and party strategists say that Mr. Ossoff’s campaign has already served as a galvanizing force, spurring small donors into action and focusing the attention of voters and activists on the battle for the House. The notion that Mr. Price’s once-safe seat could be in play, strategists said, has helped encourage Democrats in other conservative-leaning seats.