I had a weird problem where there was a short in the bulkhead connector that brings the wires out of the trunk to the tail lights. If I stepped on the brakes, voltage would sneak back through the headlight switch and activate fog lamps and make the blinkers go crazy too. So maybe when you turn on the headlights, voltage from the tail light circuit is sneaking back up through the turn signal circuits.

just a thought to check out.