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In Tennessee, state Rep. Joe Carr accused Sen. Lamar Alexander of violating conservative principles by voting to raise the debt ceiling without any accompanying spending cuts.

And in Louisville, Ky., Matt Bevin, a wealthy businessman, running against Senate Minority Leader Mitch McConnell, praised Mr. Cruz as the new face of the conservative movement and blasted Mr. McConnell, who wrote the final debt deal with his Democratic counterpart, Majority Leader Harry Reid.

“This shutdown was completely avoidable if we had real leadership in Washington,” Mr. Bevin said in a Web video. “Instead, we have career politicians like Mitch McConnell and Harry Reid that make decisions based on what is political expediency for them, not on principles and not on what is in the best interest of the American people.”

Former Alaska Gov. Sarah Palin, the party’s 2008 vice president candidate, also weighed in, urging her supporters via Facebook not to “be discouraged by the shenanigans of D.C.’s permanent political class today.”

“We’re going to shake things up in 2014,” Mrs. Palin said. “Rest well tonight, for soon we must focus on important House and Senate races. Let’s start with Kentucky — which happens to be awfully close to South Carolina, Tennessee, and Mississippi — from sea to shining sea we will not give up. We’ve only just begun to fight.”

The tea party, though, could face an uphill battle in the 2014 campaign season.

A new Pew Research Center poll that found the popularity of the tea party — including within Republican ranks — has plummeted in recent months and even more so since 2010 when it helped the GOP take over the House.

Now 49 percent of the public has an unfavorable view of the tea party, compared to 30 percent who hold a favorable view.

“The tea party’s favorability rating has fallen across most groups since June, but the decline has been particularly dramatic among moderate and liberal Republicans,” the poll analysis says. “In the current survey, just 27 percent of moderate and liberal Republicans have a favorable impression of the tea party, down from 46 percent in June.”

Jim Manley, a Democratic strategist, said that the budget stalemate may have weakened the tea party within Congress, though it hasn’t been eliminated.

“What’s interesting is that more and more senators have had it with the antics of Sens. Cruz and Lee,” Mr. Manley said. “The ‘just say no’ crowd is down to only 18 in the Senate Republican caucus, a tremendous change from only a few months ago.”