To answer the question, yes that mini-relay controls only the fog lamp circuit.
Schematic from the ford CD:
That white relay is the long rectangle box. Normally the power wire to
trigger that relay requires the low beam to be on and the high beam to be off..that's the stuff on the top left. And finally for the fog light switch to be closed (bottom left) to complete the circuit to trigger the relay. When you do the fog lamp mod all you do is connect 12V pos (top right) to where it goes into the relay on the top left.
Anyway, i guess it's possible the relay failed somehow, but i'd suggest double checking the wiring for the fog lamp mod make sure everything is secure and tight.
Edit...and i'm pretty sure you can get those little relays from some parts stores if needed.
