Two days after former President Donald J. Trump used a demeaning nickname to describe Ron DeSantis to a packed hall of Iowa Republican activists, Mr. DeSantis pointedly invoked the federal indictment against his chief rival, saying that if Mr. Trump had “drained the swamp like he promised,” then he probably “wouldn’t be in the mess that he’s in right now.”

Speaking to reporters on Sunday after a campaign event in New Hampshire, Mr. DeSantis, the governor of Florida, added that Mr. Trump’s use of “juvenile insults” served as a reminder of “why there are so many millions of voters who will never vote for him going forward.”

Mr. DeSantis has generally not used Mr. Trump’s legal troubles against him, and has instead focused on criticizing the Biden administration for what he terms the “weaponization” of federal law enforcement.

But as Mr. DeSantis seeks to reset his ailing campaign by cutting staff and organizing more informal events in the face of a fund-raising shortfall, his comments suggest he may be taking a less timid approach against the man who leads him in national polls by a wide margin. Even allies have said that his campaign has lacked a coherent message about why voters should choose him over Mr. Trump.

Part of the shift may also be a result of how Mr. DeSantis has changed his campaign tactics in the past week. Whereas he previously engaged with voters in more controlled environments, and kept the press at arm’s length, he is now regularly taking questions from both everyday Americans and reporters — meaning that he will be asked more often about Mr. Trump, who is dominating the Republican primary race.

Mr. DeSantis’s campaign reboot took him on a bus tour through rural Iowa last week. On Friday, he and a dozen other Republican presidential candidates, including the former president, took turns addressing a dinner hosted by the Republican Party of Iowa. With Mr. DeSantis ensconced in a hospitality suite not far from the main stage, Mr. Trump mockingly referred to his rival as “DeSanctis” (short for “DeSanctimonious”) and bragged about his lead in the polls.

On Sunday, Mr. DeSantis appeared at a barbecue in Rye, N.H., co-hosted by former Senator Scott Brown of Massachusetts, who served as an ambassador in the Trump administration. Mr. Brown, who is staying neutral in the race for now, is hosting similar events for several Republican candidates, although Mr. Trump’s camp has not yet reached out about attending, Mr. Brown said.

As is his normal practice, Mr. DeSantis did not mention Mr. Trump in his stump speech. But in a question-and-answer session afterward, one voter asked the governor, “Given Trump’s stronghold on what seems to be a majority of the party, what’s your strategy to show Trump supporters that you’re a better alternative?”

Mr. DeSantis responded by saying he believed many Republicans were open to nominating someone other than Mr. Trump.

“I think with me, you know, I’m the candidate that’s more likely to beat Biden,” he said. “I’m more reliable on policy. I think you’ve seen my record in Florida, and I’m much more likely to actually get all this stuff done.”

“We ended the presidency with Fauci running the government,” Mr. DeSantis continued, referring to Dr. Anthony S. Fauci, the federal government’s former top infectious disease expert, who is deeply unpopular with Republican voters. “That’s not draining the swamp.”

Hank Bivins, the voter who asked the question, said the response left him somewhat underwhelmed.

“He has to differentiate himself more,” said Mr. Bivins, 53, who is still undecided. “He’s going to have to fine-tune that answer.”

Steven Cheung, a spokesman for Mr. Trump, accused Mr. DeSantis of being “nothing more than an off-brand, bootleg version of America First.”

“No matter how much time he spends cosplaying as President Trump, he will never be him or achieve a hundredth of what was achieved during the Trump administration,” Mr. Cheung said in a statement.

A recent University of New Hampshire poll showed Mr. Trump leading the field in the state with 37 percent of the vote, followed by Mr. DeSantis with 23 percent.

But Mr. Brown said Mr. DeSantis was doing the right things to close the gap, saying that of all the candidates in the race, only field workers representing Mr. DeSantis had knocked on his door so far.

And he said that Mr. DeSantis had improved noticeably as a retail politician since he last saw the governor campaign in New Hampshire in June.

“I see him today and he’s way better,” Mr. Brown said. “And he’s connecting better.”

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