The Republicans have little margin of error. A new analysis by Cook Political showed how difficult the math was. The party would need to win five of the six Senate seats considered most competitive to recapture a majority. All six of those seats are in states that Mitt Romney won in 2012, but as Mr. Cook put it, “The path is very, very, very narrow.”

Nowhere are Republican candidates more saddled with baggage from their day jobs in the House than in Georgia, where three members are among those vying for the Senate nomination. Two of the candidates, Representatives Phil Gingrey and Paul Broun, have among the most conservative voting records in the House, and the third, Representative Jack Kingston, a senior member of the Appropriations Committee, has a reputation for being far less ideological.

In an interview last week, Mr. Kingston objected to his opponents’ premise that the primary fight was forcing him to be more rigid about the shutdown than he ordinarily would be. He said he was not even remotely centrist.

“Wait a minute, wait a minute,” he said with a wide grin on his face as he checked off the conservative organizations that had given him high marks. “One hundred percent National Right to Life. A-plus N.R.A. Ninety-five percent lifetime with the American Conservative Union. And I’m the moderate?”

He said he would side with his more conservative colleagues to defund the health care law or shut the government even if he were not running for the Senate. “To the degree that going through short-term sacrifice to change the long-term spending pattern of America, it seems to be the only way to get things done in this environment,” he said.

A fourth Senate candidate, Karen Handel, a former Georgia secretary of state, is trying to use her distance from Washington Republicans as an advantage. In a new ad, she notes that her three opponents have a combined four decades in Washington. “Their experience is expensive, and the truth is they are the problem,” she says.

In Louisiana and Montana, which have a history of electing moderate Democrats to the Senate, the likely Republican nominees are House members who have aligned themselves with the conservatives who are vowing to keep the government closed until Mr. Obama agrees to changes in the health care law.