Tail Light Issues

I would say you have two problems

1--Right socket is not grounded. This explains some of that action

2--I think you have a bad signal switch

Before condemming it, move up to the kick panel connector for the rear harness, and back probe the connector under various conditions for joltage.

If you show passable (11.5, etc is low but passable for this test) voltage, at the tail terminal, and at left/ right with the brake pedal depressed, and voltage, flashing or not, in left or right turn, you can look to harness/ socket problems

Here's how an ungrounded socket works. Let's say you have a completely normal system, and all you do is unground the socket.

With head/ tail lights off, and you either apply brakes or signal, the juice goes through the signal filament, and cannot get to ground, so it goes "in series" through the (unpowered) tail filament, and because all the rest of the park/ tail/ marker lamps on the car are also unpowered (but grounded) the "other end" of the tail filament now feeds power to all the rest of the bulbs, and through those filaments to ground.

So the bulb lights, somewhat dimly.

If you simply turn on the park/ head lights, NOW you have tail light power on the tail light filament, so ungrounded socket "bulb" goes out.

This is because you have +12 on one side of the signal filament, through the filament, through the tail filament, and +12 on the other end of the tail filament --no current flow.

With tail/ head "on" and no brakes or signal applied, the path is reversed. In this case, the only thing "to ground" is the other rear signal, because with the signal lever centered, both rear lamps are connected together to function as brake lamps.

So now power comes to the tail filament, through the filament, through that bulb's signal filament, up front to the signal switch, through the switch to the opposite side wire, back to the rear to the opposite side signal filament, through that filament, and finally to ground.

The REALLY interesting cars used to be the 60's Chevys, which had some of the lamps in the quarters, on some in the trunk LID. You really had to 'pay tenntion" to get everything properly grounded, and it used to be comedic funny to see how many of those "weren't working"