I have a 2008 Base V6 that did not come with fog lights. I purchased aftermarket fog lights with the LED halo ring (with the grill) and have tried to wire them straight into the head lights. I have the LED Halo rings wire directly into the Hi beam and they work fine. The problem comes with the fog light. I have wired them directly into the low beam head lights. They will work for a few minutes then the lights go out. You can turn the light switch off and back on and the will come on again for a few minutes then go off.
These came with a wiring harness that had a separate switch. I originally removed the switch and relay directly to low beam wire. I had the same problem.
Did you follow the instructions for installation of those lights? I'm guessing its very odd they would have you wire them that way. Basically you are overdriving your low beams and I think eventually the high current draw is popping a relay.
You definitely need to check those instructions. When I installed my Westin aftermarket lights they said to wire them to the parking lights and use the supplied dedicated switch and relay. Mine work great, and I. Your case only the LED halos should come on when you turn on parking lights, then you hit the switch to power on the driving lights. When you turn off your headlight switch everything should go off properly.
Don't leave that stuff wired the way you have it....it's just most likely wrong.
Location: PA to KY ('07) to IL ('09) to MS ('10) to FL ('11)
Never add that much load to any circuit without determining what the impact will be on any associated wiring, relay(s), switch(es), fuse(s), etc. Ford (or most any other car manufacturer) doesn't design their systems robust enough to take that kind of abuse.