Dunno 'bout the relay, but check the fuses, as well. Gauge lights and taillights are usually on the same circuit.
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1992 Deep Emerald Green Mustang GT Hatch, 5 speed, 2.73s, Mac fenderwell cold air intake, '93 Cobra MAF, Mac 70mm throttle body & spacer, Explorer intakes, Mac unequal headers & offroad pipe, Magnaflow catback, Walbro 110lph fuel pump, Jet adjustable FPR, MSD Blaster coil, Accel 8mm wires, FRPP aluminum quadrant, UPR firewall adjuster, stock cable, Granatelli upper control arms.
Waiting to go on: Granatelli lower control arms, and GT40 Iron heads!
my 95 did the same thing, it was the plastic wiring holder that the headlight switch plugs into, it had melted and let the wires touch, I had to fix the plastic piece, now all is fine.