do you possibly have a digital multimeter? If you do and it has a way to check continuity, usually where the meter will beep if there is continuity present, check that on the light switch., either out of the tranny in your hand or in the tranny with the truck in reverse and the key on, but of course the truck not running.
My dad had the same exact problem on his r150F tranny. His problem was that there was oil in the pigtail connectors contacts like marlin mentioned.
We sprayed it out with brake cleaner, cleaned it all out, and it works perfect now.
It might even be a possibility that the new switch is defective. If there is no continuity on the switch with it out of the tranny depressed in your hand, it is very possible that its defective.
Other than that, the back up lights, I think, are part of the tail light circuit and there is no relay. Ant the fuse I think is shared with the tail light fuse.
Hope this helps
