“You’re not supposed to live and die by these elections,” Mr. Sterling said, noting that in a healthy democracy, the “normal” number of death threats directed at an official like him would be “zero.” He and Mr. Raffensperger were sitting in a tavern near the Georgia Capitol early this month, monitored by a security detail. They were unwinding after another day of pitched political battle in which the Republican-controlled legislature passed an election bill that would create a raft of new ballot restrictions.

Republicans are now worried that their slipping grip on Georgia could make it a perennial swing state. Mr. Chambliss said that white suburban women, who have been the key component of the state’s Republican coalition, had defected en masse in recent years, more drastically around Atlanta than in other growing metropolitan areas around the country.

“The animosity toward Trump is real, and that’s a group that Republicans need to be courting in a heavy way,” Mr. Chambliss said. He added that such a goal would not be easy to achieve as long as Mr. Trump kept involving himself in the state’s politics.