Yep, I would look at the starter solenoid. Once the contacts overheat and burn, they're more prone to welding themselves together, causing the starter to run continuously. Burning of the contacts is usually caused by long periods of cranking during a no-start problem, one of the reasons they tell us to crank in shorter bursts with a pause in between, to let it all cool down..
A dragging starter can pull too many amps and weld the solenoid contacts also, but you would notice the engine turning over slower than normal, so that's not likely the problem here.
Last edited by Moccasin Ranger; 10-16-2019 at 01:10 PM.