I have a 1995 Dodge Ram 2500 conversion van, and for the past few months, the backup/reverse lights stay on when I'm driving. I can't find any answers in the Chiltons manual. This is the only electrical problem in the van. It has a u-haul hitch connection, don't know if that would cause a problem. I'm not sure where to start looking.
Wow! Takes me back to the old gearhead days. Reverse lights on while driving was a sure sign of a failed switch associated with the floor shifter.
A quick look at a wiring diagram for your truck shows that the back up light control is part of a neutral safety/backup light combination switch. Either it is failing or you have a short on that circuit to a " HOT" wire.