Mr. Rispone is a Baton Rouge native with his own bootstrapping Louisiana story to tell: In 1989 he and his brother founded ISC Constructors, a specialty contracting company that now boasts annual revenues of $350 million, according to the campaign.

Mr. Rispone has said he would follow the Trumpian model for economic growth, with a focus on cutting taxes and regulations, ideas with a natural constituency in a state that is dependent on the oil and gas industry and wary of big government.

This heavy emphasis on tax cuts is at the center of one of the few actual policy debates in the race. Mr. Rispone has argued it would unleash business growth, but Mr. Edwards has said it would return Louisiana to the days of slashed services and budget shortfalls brought on in part by the tax-cutting fervor of Bobby Jindal, the former governor whose fiscal troubles helped bring a Democrat like him into office in the first place.

Mr. Rispone’s critics have said his embrace of Mr. Trump is a fill-in for his lack of policy details beyond the general talk of cuts. This week, the radio host Newell Normand, a former Jefferson Parish sheriff, blasted Mr. Rispone on his influential New Orleans radio talk show.

“Eddie Rispone’s strategy thus far is say nothing, reveal little, ride the coattails of Trump and therefore wait to see if Trump can drag him across the finish line,” Mr. Normand said. “All the while Eddie Rispone has already sold us out — we’re actually now, in nationalizing this governor’s race, a pawn in Washington politics. I don’t know about you, but as a Republican, I find that very distasteful.”

With early voting already underway, Louisianans, like the rest of the country, are wondering exactly how much stroke the president still has. Mr. Trump held rallies in Kentucky and Mississippi before this week’s governors’ contests in those states, with what appear to be mixed results: Matt Bevin, Kentucky’s Republican governor who had angered voters with his derisive comments about protesting teachers, came about 5,000 votes short of his Democratic rival, Attorney General Andy Beshear. Mr. Bevin has not conceded and a recanvass of votes is scheduled for next week. But Mr. Trump’s favored candidate in the Mississippi race, Lt. Gov. Tate Reeves, handily defeated a popular Democrat, Attorney General Jim Hood.