The plug is a big red square plug, on the passenger side, IIRC It joins the smaller harness to the rear of the car to the main engine bay harness.
My pen is pointing at it in the two attached pictures.
IIRC - isn't there some terminal "eyes" on each of the taillight buckets? I am sure you have checked those as well?
It still sounds crazy that the Turn Signal and Hazard Flasher behave differently... maybe the circuits ground differently.
Rocky