Jeff,
I believe (not positive) that all of the lamp sockets in each lamp housing share a common ground (a grounding strap), so if the running & brake lights are working, this indicates it is not a ground problem.
You've mentioned several times that with the trans in reverse you have power on both sides of the reverse switch. This tells me 2 things, your fuse is OK and the switch is OK. Lets move on.
With the key on & the trans in reverse, just the way you tested the back up switch, make sure your test light is grounded very well, using it's pointed end, pierce the insulation of the wire going to the back up lamps at the socket, do you have voltage there?
If not, you either have a broken wire between the back up switch and the lights, OR the wrong wire.
George
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