1998 Plymouth Breeze Expresso

I bought this from a friend who informed me the starter needed to be replaced. Upon replacing the old with a new one, the car started and seemed to be running fine. I drove it, brought it home then shut it off. I tried starting it again and nothing. I popped the hood and examined the starter finding the solinoid was hot to the touch. I assumed the starter was a dud so I took it back and got another. Same problem. This time I noticed the starter was staying engaged with the flywheel. There was no shim present when I removed the old starter so I assume no shim is needed for this vehicle. Ive replaced the black ignition box located opposite the lock cylinder thinking that was the problem. It wasnt. At the cost of yet another starter. Ive checked the starter relay, good. Ignition swith, good. Starter, brand new, (no refirb.) There doesnt appear to be any burnt wires. Fusible links check out fine. Ive ruled out a shorted wire causing the starter to stay engaged because if this where the case the starter would kick on simply by placing the ignition to the run position. This car has managed to completely hide the cause of this problem. There has to be an answer to this problem :) Its a nice looking, good running car that id rather not give up on. Any help would be greatly appriciated. Thank you.