I am having same issue with my 99 f350 . So I'm hoping you can tell me what was your fix?

 

I have a 2011 F150 that had the same problem and after check all fuse and relay, just found when I check in the top of the spare tire found a wires bite for a squirrel I hope this problem give some ideas. Fix the broken wire and all working fine.

This wires was broken by squirrel.

 

Ok.. so I went down same path with my 2014 f250.
checked all fuses relays etc.
My tail lights would blink with key fob but not go on with headlights,.
Finally i looked at bulbs. They appear to be ok however when i checked for power with either bulb in place i got no power.'
When i pulled both bulbs I got power out of pig tail. (short in light bulb)
Yup. Replaced both bulbs and it worked.
The lights that blink with keyfob are driven by the security system.
The tail light relay is controlled by the body control module. The control module sensed a short and kept them from lighting.

Hope this helps guys