Engine will not shut off

This is a fuel injected (non carburetor) 350 CI engine correct?

If that is the case, the only way this could continue to run when the key is turned off is if the PCM is continuing to command fuel and spark. That is only possible if the PCM is continuing to get power or the ignition switch is faulty. The fact that the lights shut off points to a power issue that is back feeding to the PCM.

Have you changed anything or can you see if it look like there are any aftermarket parts with the ignition system? Is the alternator new?

I have seen Alternators cause this issue so just unplug the connector on the alternator (not the B+ power wire). Basically you want to unhook the PCM from the alternator and you do this by removing the connector (number 2 in the attachment).

Let me know if it keeps running with this unplugged and the ignition is off and we can move onto something else.