I'm not entirely sure, but I feel like they're separate issues.
I'd probably service the transfer case and make sure there's not any glitter sauce (or chunks) in there. Fresh fluid couldn't hurt either way. I'm not overly familiar with the 246 AutoTrac cases, but they're pretty similar the the 263s. There's the actuator on the front diff and an encoder (shift) motor on the side of the transfer case. Assuming the internals are good, none of the fuses are blown, and there's no TCCM codes, I'd have to lean towards the encoder motor. I'd check for codes first. You'd need a pretty decent scanner to see the codes. Off the top of my head, they'd probably be "U" codes. Similar to BCM fault codes, they usually point toward communication errors.
As for the backup lights... does the dash display properly when you shift through the gears? No blown fuse? (LP/BU and Veh BU) Bulbs good? All the other functions of the tail lights work? You can likely find the pinout for the NSBU plugs and check for power there. It may be in the wiring between the pigtails/NSBU and the reverse light socket itself. If I remember right, that wiring bundle is ground to the same crossmember as the spare tire hoist. The ground could be rough.
