I determined that everything worked fine, AS LONG AS I held the START button down, so it had to be a safety switch somewhere that caused the problem.
I removed the lid and front, then began to test each safety switch that I could find. After determining that none of the visible switches were bad, I had to remove the drum and found a few more switches down around the motor.
After checking them, I was able to determine that the switch at the idler pulley, the pulley that keeps the belt taught, was bad. I did a temp jumper, effectively bypassing the switch, and used the dryer until the new switch arrived.