Sounds like a tough one, does the car have twilight
[ zone ] headlights that stay on after car is turned off,
or other features like autotronic dimming. Are you sure only one side
lights, or does one side glow brightly, and other side barely glows ? I
had a nasty problem with my 81 El Camino that turned out to be a shorted
marker light socket. Found problem when I cut wires in disgust to make,
still another, test light. If I hadn't lost my patience, I may * never *
have found problem.
Could also be funky directional switch, but even the
worst ones going shut off with key. I hope it's not the printed circut !