PARIS, Tenn.—Republican Don Sundquist, a onetime governor of Tennessee, has a history of feuding with Marsha Blackburn, a GOP state politician turned U.S. congresswoman. Last month, Mr. Sundquist was spooked by signs a Democrat could win in his red state and decided to bury the hatchet and endorse Ms. Blackburn, who is now running for the Senate.

“We need to keep that seat Republican,” Mr. Sundquist said in an interview. “Some things are more important in politics than personal feelings.”

Republicans...