Why would the Starter engage before turning the key to start position and then the starter stay engaged while engine is running. Burned out the starter

Check the routing for your ignition remote wire (the small one that attaches to the starter) make sure it is not receiving power in the "ON" position only in the Start position. This can be done with a multimeter in the "Resistance" position.

1) Disconnect the Negative Battery cable from the battery
2) remove the starter from the vehicle
3) attach a multimeter lead to the negative battery cable
4) have someone assist and turn the key to the "ON" position
5) connect the other multimeter lead to the starter ignition wire (the thin/small one)
6)if you have a reading of "OL" then you are fine, if it reads "0.00" or higher then you have a short or that wire is powered at the Wrong position on the key.
7)if you did get a reading locate where the wire connects on the ignition harness by putting one lead of the multimeter onto the ignition lead and then testing each lead in the ignition switch harness (firewall side not key switch side) you should get a "0.00" reading. if you get a hit on more than one contact on the harness you have a short.

Inform me if you resolve this issue this way, if not there are more steps to troubleshooting you can use and i can walk you through it.

I would be looking at the solenoid, too.