Do not buy any parts just yet,, we need to isolate this out a bit.
It will be pretty simple to figure out which direction to look..
Once it is started, turn it off via the start/stop switch. If I ready you correctly, the starter will still be cranking.
Unplug the small wire (start wire) at the solenoid. If it keeps spinning it is a sticking solenoid which is very common in the automotive world. If it stops then it is in the start circuit. Probably not the switch as you can turn it on and off via the switch. So I am guessingFor now, a sticking relay.
For now, do the little test I just explained and we can go from there..