So when you switch the key to the run position, not the crank position, the starter engages? Does that model have a starter solenoid on the starter or is it remote? Unplug ignition switch and see if the connector has melted 2 pins together in the connector, if it has a start relay it could be stuck closed possibly....
Ok so the start relay maybe stuck.....im sure youve checked this....but on that same circuit there is a diode that supplies ground which maybe supplying the ground all the time...