You are thinking backwards, bad ballast will start then shut down when the key is released. The problem here sounds like a lack of voltage to the coil when the key is in the start position. Then once the key is released and returns to the run position, current is supplied to the circuit. Crawl under the dash and check the wiring to the ignition switch and make sure everything is hooked up. If so, you will need to trace the wiring. Pay particular attention to the bulkhead connector
I'm with Hemi Rebel on this one. Electrical issues drive me batty too, but I had the same symptom on my 360ci motor last year. Actually, the car is in the body shop and still has the problem because I never got around to trouble-shooting completely and fixing it, but I know my issue was NOT the ballast resistor alone. My troubles started when I replaced parts as follows:
new stock replacement coil
new 1 ohm 2 prong ballast resistor
new cap and rotor
new solid state voltage regulator
original mopar chrome box control module remains installed
new replacement control module wiring harness (can't remember if it's 4 or 5 lead)
rebuilt mini-starter
Somewhere on another forum I read that this issue is caused by not enough spark and when that is combined with the load caused by engaging the starter, the motor won't fire. Then when the load is released, the voltage shoots back up and the motor turns over. If it does fire when cranking, then there may be a mismatch between the ballast resistor and one of the above. I heard that the 2 prong type ballast resistor requires one type of control module and the 4 prong type requires another type. Am I remembering this correctly?
They said to trouble-shoot and see if this is the issue, run a wire lead directly from the positive side of the coil to the positive battery terminal and see if the motor fires when cranking.
Also, like Hemi-rebel says, check the bulkhead connector in the engine bay and also under the dash for other wires to rule out a short or loose connection. I know from personal experience that sometimes wires can also sag down and if the pedal linkage is pressing on them you can loosen things up or fray them. Make sure all of the wires are neat and nicely tucked under there. Zip tie if needed.