Gov. Bobby Jindal of Louisiana seems equally worried about upsetting the party’s base; he suffered blowback from Republican activists after delivering a stinging critique of his party in the wake of the 2012 presidential defeat, saying Republicans must avoid being seen as protecting the rich and should “stop being the stupid party.”

Now he is more measured. He declined in August to join a bipartisan group from the National Governors Association that warned Congress not to shut down the federal government, and last week he told reporters he would not “micromanage what’s going on on the Hill or second-guess tactics.”

Gov. Chris Christie of New Jersey is seizing on the moment to distance himself from Republicans in Washington and its dysfunction. The governor, who is up for re-election next month, fires off near daily Twitter posts about his bipartisan achievements in his home state with the anti-Washington hashtag of #DearDC.

His aides sent reporters a link to a video in which he announces, “I hope in Washington what they figure out is that what we pay them to do when we send them down there is to run the government, not to shut it down.”

Gov. Scott Walker of Wisconsin has been more restrained, avoiding a hashtag war, but has sounded similar notes, while former Gov. Jeb Bush of Florida appears to be assuming the role of the adult in the room, warning his party not to overreach, because of what he called the “dicey” politics.

“It’s embarrassing to see this play out as it has,” Mr. Bush said. “And I respect, there’s deep — deeply held views on this. But you’ve got to pay your debts.”