LAURINBURG, N.C.—In this impoverished stretch of North Carolina, the clock is ticking on President Trump’s promise of economic relief. 

Across the pine forests and cotton fields of Scotland County, the signs of loss are everywhere. Medical-device and towel factories shut down some two decades ago. Boarded-up brick homes blight residential streets in Laurinburg, the county seat. When a high-school teacher asked his students to imagine where they would be in five years, some answers were bleak.