The inherent higher circuit resistance of the LED bulbs in the headlights changes the current sensing of the headlight/fog lighting circuit. As a quick workaround to fix the problem I did the following on a 2000 Ram 2500 Diesel with LED headlights and turn signals:
1) Locate the Fog lamp relay in the main fuse/relay compartment under the hood. Mine only has 1 fog lamp relay located at the bottom of the box closest to the fender.
2) Remove the relay and put in the spare slot or keep elsewhere for future needs.
3) This is a SPDT type relay with a "Normally Closed" and a "Normally Open" contact. The "Normally Open" contact is not used. The "Normally Closed" contact is what controls the fog lamps in conjunction w/ the Hi/Lo headlight switch.
4) Make a small 2" jumper using #16 or larger insulated wire and two crimp male spade lugs. These should be the same size as the relay terminals.
5a) Insert the jumper between terminal 3 and 5 in the socket where the relay was plugged in originally. This will permanently close the "Normally Closed" circuit. Be careful and get the right terminals in the socket. Refer to the diagram on the relay and the terminal numbers on the bottom of the relay.
5b) As an alternative, you can snip the #1 relay terminal on the base of the relay, thereby disabling the relay, which would accomplish the same thing. You would not need the jumper above if you do this option. Make sure you plug the relay back in if you do this option as the relay provides necessary continuity to terminals 3 & 5.
After doing either 5a or 5b above, the fog lamps will work from the fog lamp switch on the dash regardless of the headlight Hi/Lo switch position. So, the fogs will work in Parking only, Lo Beam, and Hi Beam and be controlled solely from the fog lamp switch.
Hope this helps.