Starter Run on

When the starter runs after the engine starts, is there voltage at the little push on stud? If so you either have a bad ignition switch or a wiring problem.

No voltage at the little push on stud, check for voltage at the big stud on the starter solenoid that has the wire that goes to the starter. Voltage here, and the starter solenoid is bad.

No voltage at the little stud or big stud that has the starter power feed wire, then the starter has mechanical problems and is hanging up. Pull the starter & look for damaged teeth on the gears, bent parts or other mechanical problems. Plan on replacing the starter, but also plan on fixing any other problems you find along the way.