I agree with Al. You could let her run until she dies and then check for spark.
The engine shutting down like the key was shut off is indicative of an electrical failure of connection. The "fact" that it starts up after "cooling" down a bit and then dies again is a classic coil failure.
Check some of the threads regarding the coil issues and don't get caught up in the technical stuff. You will need a new coil and a resistor for it. My suggestion is buying them from our host or Indigo and be done with it.
Dave Neptune