Added to the mystery is this:
With the engine running and headlights on I turned the key switch to the off position and the headlights stayed on. Which is desired, but not what has been happening so I thought for some reason the problem amazingly corrected itself. However, when I turned the headlights Off, and then then turned them back ON they did NOT come back on.
I just don't understand what could have changed to make the headlights not come on when the engine is not running, and now STAY on when the engine is turned off until the headlight switch is turned off, but then will not come back on immediately (engine off) until the engine is started again.