Sorry for the deleted posts. I've got a nice update.
CJJeep, you ended up being correct. My old switch was snapped in half, which made it look like it wasn't the switch at all. I took a wrench to it and removed it, here it is next to the new switch:
Once that was done, I was tested the switch by putting my meter on the leads (continuity) while in neutral - no contact. Put it in reverse and I had my closure. Switch is in and working!
Next I was able to confirm that my back up lights were functioning by touching the red wire (which should have 12 volts referenced to ground) to the white wire. With that contact made, my reverse lights lit up.
*John, I think they may have been working all along. The problem is that with my ORO LED tail lights, there is hardly any white light when in reverse. When I was testing from inside the engine bay, I was expecting to see the lights turn totally white - but with these, the red stays on and you get like 5 little LEDs to show white. Better than nothing!
While I was at it, I tested the 4WD indicator (orange wire). Shorted it to ground and the dash light indicator came on brightly!
Last step is to solder on the female end of the harness connector to made my plug sides to my harness sides and life is good!
Thanks, as always, for the input guys!