Wow, the first pic is interesting. That bulb was arcing for power. That's why the connector is black.
Since it works when the headlamps are off, I suspect we have a ground issue. A weak ground can act this way when loaded.
I attached the entire wiring schematic for the exterior lighting. It was three pages long, but I had to cut each pic in half to make it readable. I did overlap them so you can follow from one to the next.
Now, there is a junction block for the grounds related to all taillights. See pic 4. Note that all of the ground wires are black in color. Try running a jumper wire from the right-side light sockets to a known good ground to see if that makes a difference.
Also, take a test lamp and see if (with the headlamps on) there is power to the socket for the signal. Make sure the test lamp has a good ground also. That way, the light itself will provide a ground path.
Here is a link you may find helpful:
https://www.2carpros.com/articles/how-to-check-wiring
Let me know what you find or if you have other questions.
Take care,
Joe
See pics below.
Images (Click to make bigger)
Sunday, October 17th, 2021 AT 9:39 PM