“Our challenge is not just to replace Obama in 2012,” Ms. Palin said. “But the real challenge is who and what we will replace him with?”

At the same time, there remains a debate within the party about whether the candidates are moving too far right in pursuit of their most ideologically committed voters, and about how to balance principles against the assumption that winning the White House requires the ability to appeal to centrist voters. In recent weeks, former Gov. Jon M. Huntsman Jr. of Utah has been making the case for moderation, attracting new attention.

“Republicans have to nominate someone better than the person they want to defeat,” said Mike Huckabee, a former Arkansas governor and 2008 presidential candidate. “If they get so adamant that they will only support a candidate that believes everything on their checklist, they will re-elect Obama.”

There are interwoven crosscurrents coursing through the field, but Mr. Perry, who has reshaped the race since declaring his candidacy three weeks ago, is at the center of nearly every Republican strategy.

He is courting fund-raisers and reassuring potential supporters that he can withstand the intense scrutiny accompanying his new prominence, even as he scrambles to drain establishment support from Mr. Romney and erode enthusiasm for Mrs. Bachmann among Christian conservatives.

He invited a group of contributors to the Barton Creek Resort and Spa outside Austin, Tex., last week for a political briefing and dinner. One participant said the governor reassured his guests that he “could take a punch and land a punch” and survive the glare of a presidential spotlight.

He convened a separate meeting with evangelical leaders, where attendees told The Texas Tribune that Mr. Perry said, “There is nothing in my life that will embarrass you if you decide to support me for president.”