Right rear turn signal will not flicker. New bulb. With headlights on. Bulb does
Light up, but will not flicker. With headlights off directional bulb will not light at all.

Sounds like it is your ground wire. Follow the black wire and make sure it is grounded. Could possibly be attached with a screw some place on chasis.

I hate to break this to you...
The ground wire is White with Black trace: W-B
And the problem is likely the socket shorted internally.

5171

Does sound like ground issue, can be a body connection or at a socket