“Of course there’s politics. Of course it’s become a partisan fight, but I try not to look at it like that,” said one of them, Representative Patrick Murphy, a Florida Democrat who won his seat in 2012 against a Tea Party favorite, Allen West, by fewer than 2,000 votes. “I would imagine there will be more and more attacks coming for months on end.”

Republicans say the travails of the health law are not about to let up. Even if the website is fixed, more shocks will follow, they predict, maybe in the form of data breaches or anger over the tax penalties in the law for those who do not sign up for medical coverage. The problems — with shrinking provider networks, limited competition, rising costs and layers of regulations — are so complex that they simply cannot be fixed, Mr. Sanborn said.

For example, Anthem dropped not only Concord’s hospital from its network but also Laconia’s oncologists and cardiologists. Cancer patients in Laconia who purchase health plans through the health law will have to drive past two hospitals to get their chemotherapy in Manchester, Mr. Sanborn said.

For Democrats, voting for Republican health care bills may not be a political panacea. After Friday’s vote, the National Republican Congressional Committee mocked vulnerable Democrats who voted yes as political turncoats running from past votes. Ms. Shea-Porter, the New Hampshire congresswoman, said she understood that — and had no intention of playing down her support for the health law, which cost her a House seat in the 2010 wave before she won it back in 2012.