From a PM I sent Noles:

It ain't a relay. See page 8W - 10 -18 of the (Non-Srt) Service manual.

The headlamp switch provides power directly to the lights, no relay is involved. (Sorry, we all hoped it was a relay). To make the lights come on for that "Courtesy" thing I spoke about, the "Central Locking Pump/Security System Module" feeds power into the light switch on a pin that bypasses the switch - this feeds power directly to the light bulbs.

So... it has to be the above-mentioned module! What I would do:

1) Replace that old battery, as that makes the modules act up.

2) But as a test, pull the negative battery cable off the battery and let the car sit for at least 2 minutes. Put the cable back on and start the car, turn lights on - move the car 5 feet and park it, turn the lights off and engine off. See if the lights come back on when you exit the vehilce, then turn off after 90 seconds or so.... which they are supposed to!

What I am getting at is:
MAYBE the module is just confused, probably by low/unstable voltage due to an old battery. Old batteries in these cars can make the radio act up, the car not start, and some accessories take on a mind of their own, etc.

See what happens after pulling the batttery cable off for a bit. But again, ANY battery over 3 years old is a potential chance to walk home or call a tow truck!

My new battery was under $93 with all taxes at O'Reilly auto parts.