Since you stated the bulb socket is okay, then you still have a ground issue.
I'll use another example to explain how this could be. The fender side marker lights do not have a two filament bulb but they still blink with the turn signals when everything is correct. They do not have a dedicated ground. Instead they are fed with two hot wires, How does this work?
One wire feeds from the parking lamp circuit, the other feeds from the correct side turn lamp. When just the parking light circuit is on, currant feeds from the bulb filament to the signal side, then through the turn signal bulb filament to ground. Since only a small amount of amperage is needed for the 194 bulb it does not light the signal bulb.
When the parking lamps are off and you use the turn signal the side marker lights up and grounds through the rest of the parking light circuit.
When the parking lights are on and you use the turn signal. the fender marker blinks off, two positive feeds on at the same time.
Now to your issue. Since you stated the turn lamp gets brighter when the headlights are on, this can only be one of two problems. One, the headlight wire is shorted to the turn wire, or you have a ground issue. Since all of the same side front lights ground on the same bolt, one or all of the grounds is open. This allows the headlight current to light the turn lamp. Then finds whatever different path to ground.
Two last questions, when you checked the turn lamp socket you have a ground, correct? Did you turn the headlights on to see if you still have a ground, or maybe now it is positive with the headlights on?