Health care is a signature issue for Mr. Edwards, who expanded Medicaid soon after taking office in 2016. On Wednesday, he was at Baton Rouge General Medical Center celebrating a planned expansion of a burn center, the kind of progress he said was possible only after he and lawmakers were able to correct the deficit.

In an interview, Mr. Edwards, 53, said that the Democrats running elsewhere might benefit from a similar “common sense” style. He added that it was common sense to not alienate Mr. Trump, particularly in a state in regular need of federal disaster relief.

“I make sure we have a good relationship with the folks in Washington, including the president,” he said. “I’ve been there to visit with him nine times over the course of the last three years.”

Republicans have knocked Mr. Edwards for raising taxes and have accused him of creating an unfriendly business climate. Dr. Abraham, a family medical practitioner, has also run an ad declaring that there are only “two genders,” perhaps foreshadowing an attack on Mr. Edwards’s efforts to expand L.G.B.T. rights.

Mr. Edwards’s supporters are hoping he can benefit from a Republican house divided. Under Louisiana’s “jungle primary” system, all of the candidates will be on one Oct. 12 ballot, with the top two vote-getters going on to a November runoff if no candidate gets at least 50 percent in the first round. Mr. Rispone, who has been running third in the polls, recently unveiled an ad raising questions about Dr. Abraham’s fealty to Mr. Trump.

In Mississippi, Mr. Hood will have his hands full cobbling together a coalition of African-Americans, liberal white city dwellers and exurban moderates. His superpower is a full immunity to charges that he is anything close to a latte-sipping urban elite; his first TV ad shows him packing shell casings, fixing machinery next to an American flag, heading to church, sitting in his truck and riding a tractor.