engine running

no probably ign switch if not the relay. it's got
OFF - ACC - RUN - START

when in the start position it bypasses balast resistor to give full juice to the coil (ok for short time)

when you let the key go to the run position it goes through ballast resistor (partial juice to coil, normal operation)

if you are letting the key go and it dies, it's either the ballast resistor or the switch itself is jacked up