PENSACOLA, Fla.—When Donald Trump swooped into the conservative heart of this battleground state for a rally, the 10,000-seat arena sold out. Some supporters had arrived at dawn to get a good spot for Friday night’s event.

Half a mile west of the Pensacola Bay Center, the Republican nominee’s yet-to-open campaign office for the county was locked and empty. A Saturday opening was pushed back to this week.

Meanwhile,...