Sounds like you have several problems:
- The starter sticking on can only be one of two things - the ignition switch, not the lock but the switch, or the solenoid. 9.9 times out of 10 it is the solenoid. And, aftermarket solenoids are famous/notorious for sticking on - even right out of the box. The Ford solenoids, even ones from a salvage, are a much better bet.
- Hot wires means you are pulling too much current or you have a poor connection/wire. I'd bet on the poor connection or bad wire, and suggest you check the connections at the battery and on the engine. Further, check the battery cables where the wires go into the terminals as those are known to corrode and cause problems. If your connections and wires are good then your starter is dragging and, therefore, pulling too much current. Have it checked at the parts store.