Happy New Year.
The taillights on my GT were continually causing problems until I finally gave in and replaced them.
The sockets and back plates are steel, the contacts for the bulbs are mounted on a piece of fiberboard pushed upward by a spring. The whole thing grounds to the body by the 4 small pop rivets. The usual problems were weak contacts and bad grounds.
Assuming you have voltage to the turn signal side of the bulb, the socket and contacts need to be clean and free of rust and corrosion. If not the turn signal circuit will use the running light circuit as the ground. This looks like a dim or out of sync flash when the running lights are off. Really bad and it won't work at all. Running lights can also ground through the brake filiments at times.
As the backing plates rusted it became harder and harder to keep a good ground through to the body. Drilling out and re-riveting would only work for a while.
If the sockets are good you might want to consider taking off the whole assembly for cleaning. Make sure the body is clean and put a new seal in the groove on the backing before riveting it back on.
Good luck, hope this helps,
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