You didn't state this very well. You cannot have voltage at the two outside terminals with the connector off, and there wont' be anything at the transmission (in reverse) except CONTINUITY between the two outside terminals OF THE SWITCH.
As Red said, you can pull off the connector, there will be power WITH THE KEY ON at ONE outside terminal. You should be able to jumper the two outside pins of the connector together and get the lights to work.
If they do, it's a transmission/ switch problem
If they don't it's in the wiring somewhere, a bulbs problem, or a socket(s) ground problem.