Really weird electrical problem 98 Caravan

My 1998 Dodge Grand Caravan has a REALLY weird electrical problem with the driver's side turn signal. The signal works properly except when the headlights are on AND you're stepping on the brake pedal. If the headlights are on and your applying the brakes the left turn signal will flash double speed as if the signal bulb was out, but the signal is actually flashing double speed and the reverse lights on both sides flash with it! What makes it even weirder is that the bulb checks fine on a multi-meter, but if I swap it for a "new" bulb the problem disappears for a week or so. I recently replaced the left side tail light housing with a Dorman replacement part. I am completely baffled. Does anyone have any suggestions??

Sounds like a ground issue.

check the socket where it plugs into the light for corrosion or cracked insulation on the ground wire - as stumpy suggests, a bad ground.

bad ground.

I checked the plug and the housing and didn't see any damage but when I put the plug back on I pushed really hard and it seemed to seat slightly deeper than it was before. The problem has disappeared for now. Hopefully it's permanent. Thanks for all the help guys! I'm sure you're right about it being a bad ground. If the problem returns I'll do another search for the problem in the wiring.

Matt, it doesn't take much it seems to go screwy.
(The car, not you - you've been screwy for at least 35 years )
The Mighty Wheeze gets condensation in its R/R tail light, and every couple years I gotta take it apart, clean it, and the signals work well until the next go-round.
290,000 miles now, and I'm gonna see if the little guy will outlive me...