anything is possible, but not really likely that the transmission is staying so locked up that the starter burns up.
I have seen some of these have a problem with the starter after someone else changing out the key switch with one that was not OEM, and getting some of what might be called 'high side feed back' where a diode was no longer blocking a signal to the starter that came through the neutral safety switch and directional switches from/through the back up alarm. it caused an intermittent engagement of the starter while the motor was already running. (electric current looking for the easiest path to ground, found a path through the starter solenoid to to be easier than through the correct circuit.)
Are there a number of 'inline diodes' in the harness under the dash/behind the meters?
and of course, load test the battery first.
after you put on a 'known good' starter, if you disconnect the small wire that causes the starter to engage, after the motor starts up, do you still have the power cable carrying flowing current? (a clamp on DC amp meter on the power cable will help ease this part of the diagnosis).
and of course, load test the battery and check for voltage drops in the circuit to the starter.