It does not HAVE to be the switch. If I were you I would pull out the starter and hook it up and turn the key like you were going to start it and see if it stay engages when you release they key. Then remove the power wire from the start and test for power, if it still has power with key released follow the wire back. It can either be a shorted wire to power, or a relay/solenoid sticking closed or the ignition switch sticking closed. I don't think the starter could be causing it as it has to get power to spin if the power was being cut correctly it would stop spinning. or maybe its that blower fan staying on not the starter? First find out for sure what is staying engaged with the key off.