Turn Signal Relay Issue

I think I'd start by diagnosing the brake lights. In most early Mopars, the brake light filament in the bulb (1034's or 1157's) is the same one as the turn signal uses. The turn signal switch just cancels out one of the stop lights depending on which way you move the lever. The turn signal circuit depends on load for the flasher to work. Either use all incandescent and the stock flasher, or, the LED version with the LED bulbs. I wouldn't try to mix and match.