Forsure the starter solenoid is stuck in the engage position, however it could be one of two things. Either like dough said, bad solenoid, or it could also be a bad relay. Both of which are not terribly expensive and easy to replace. This is only if you are sure it "wont shut off". When you DC the batts does the teeth of the starter stay engaged with the flywheel? You should be able to isolate the starter solenoid by disconnecting the smaller wire coming from the relay, then reconnect the batts. If the same problem, then dough hit a home run. If not, then replace the relay.