Energizing Mr. Trump’s loyal base is essential to the GOP’s runoff strategy, and Republicans in Georgia and in Washington, D.C., would like to see the president train his fire on Messrs. Ossoff and Warnock. But Mr. Trump, apparently preoccupied with a continuing recount of the state’s presidential results, instead spent the past few days on Twitter attacking top Georgia Republicans: Secretary of State Brad Raffensperger and Gov. Brian Kemp, both once considered allies of Mr. Trump.

“This could really go off the rails and really cause long-term damage,” former GOP state Rep. Buzz Brockway said Saturday. “The long-term health of the GOP is on the line here in Georgia.”

More than a dozen Republican officials and strategists said they worried the intraparty feud was distracting from the runoff effort and could hurt the party’s chances in 2022, when the governor and one of the Senate seats will be on the ballot…

Rusty Paul, a former chairman of the Georgia GOP, said the senators had no choice but to publicly disavow Mr. Raffensperger because if Mr. Trump turned on them in frustration, it would be disastrous for their re-election prospects.

“If the president is tweeting bad things, the base is not coming out,” Mr. Paul said.