“My opponent has a voting record that is very much in line with Tea Party thinking,” said Representative Cheri Bustos, who defeated Mr. Schilling last year.

As for the Republicans, their focus on the health care law has shifted from talk of a repeal to criticism over its rollout.

“The health care law is going to be enormous in this race,” said Nan Hayworth, a former representative who is trying to make a comeback in upstate New York. “But this is about people’s lives and the system not working. It is not a partisan issue.”

The candidates’ approach differs from that of Republican incumbents, who hope to fend off challengers in the primary by focusing on undoing the law and on investigating the role the administration had in its flaws.

“The regular Republican repeal language would not be an asset to former members trying to come back in competitive districts,” said Nathan Gonzales, deputy editor of the nonpartisan Rothenberg Political Report.

Whether the health care law or the shutdown will help candidates remains to be seen.

While most voters blame Republicans more than Democrats for the shutdown, the scorn for all of Congress is historic in its depth. The health care law, while under fire, remains popular among Democratic voters, whom most of the returning candidates need in order to win. And even among the opponents of the law, there is a sense that dysfunctional politics underlie its problems.

“To me, the blame game is played by the whole Congress,” said Curt Snyder, a vice president of Numerical Precision, a component manufacturer in Wheeling, Ill. Mr. Dold’s opponent, Representative Brad Schneider, visited there recently to discuss the shutdown, which temporarily hobbled the company. “At least he is taking the initiative to stop by and say we can’t let this happen again.”