Actually, just thought of another thing that would cause the problem.
Even more likely than a bad relay.
There is a switch that opens when you open the door. *Its likely broken..
If you can locate it, jump the contacts (CAUTION: 117 vac here) and see
if that keeps it on, and if so, replace it.

The door switch works- that is, when i open the door it stops, when I
close it the dryer runs, as long as I keep the start button pressed.