By "running lights" I take it you mean the tail lights? They are on a separate circuit from the rear turn/brake lights altogether which is why those ones are working OK.
A further assumption is: they are all not working and you've checked the bulbs/fuse and found them OK.
One possible cause could be the headlight switch itself, although I have a feeling the problem may lie in the wiring running along the frame to the rear of the truck. Any spots where the wiring has been joined are suspect, I would suggest beginning at the headlight switch with a test light and working your way back along the circuit until you find a spot where the electricity isn't getting through. The brown wire is the one you're interested in - that's the tail light one.
Regarding the burnt wire under the dash that you found, you will have to trace it back and see where it leads to. If it is related to the tail light circuit it will come from the headlight switch.
Be patient, electrical repairs can be frustrating, I know I've used a few bad words when chasing down wiring problems..........................