Let me start this off by saying, I am not mechanically inclined by any means.
I have an 05 4.7. Haven't any electrical problems with it since I got it about a year ago.
A few weeks ago, my fog lights stopped turning off. I didn't realize it until my battery died. No matter what I do, they will not turn off until I unplug my battery.
I've had a few people tell me to go through my fuses and check those, and I did, and I'm at a loss of what else it could be besides a short in the wiring or something.

 

I have a 2004 and the fog sw is on the dash switch turn to right to put lights on and pull out for fog lights. If yours is the same it could be the light switch.

Or if yours is on the steering column multi switch it could also be that.

Both are easy to replace.

If yours is on dash like mine remove the plug on back of switch and see if the fog lights go out could be a switch or maybe a relay stuck closed.

Good luck

 

I unplugged it and they didn't turn off.
Where would I start with wiring? Or would my best bet be to get a new relay?
When I pull the relay, they shut off.

 

Look where you took that relay out there should be others with the same part #.
Swap out the relay and see if it cures your fog light problem.
If it does work then put the good relay back from the position you took it from.
Leave out the bad relay until you get a new replacement.
Most part stores should have the relay or just go to fleabay and put part # in and you should find on line. Or to the dealership for the relay.

Good luck