1981 Ford F-250 Starting

What was wrong with the old regulator? Do you have the new one bolted to the body?

When the engine won't stop running, remove the white "S" stator wire from the back of the generator and see what happens.

The problem you listed is the engine won't stop running unless the voltage regulator is unplugged. Something is keeping the ignition system powered up when the ignition switch is turned off. The stator terminal on Ford generators can do that if there is a defect somewhere else. If removing the "S" wire allows the engine to stop, we'll know that is the direction we have to go. The alternative is the new regulator is defective or plugged in backward.

What exactly was the symptom with the old regulator? Was that the taller mechanical unit or the newer electronic one? Is the new one of the same design?