No Back up Lights

Manual Transmision, 1994

need to know what you mean by checking circuit. when you installed the new switch is it making contact the way it is supposed to? if you take one of the sockets and use a volt meter you can check if you are getting power to the socket.  take the negative side of volt meter and connect it to a good ground (also could use a test lite). with the positive end touch the hot side in the socket. if it shows power then you have a bad gound on that circuit. corvettes are very finicky when it comes to grounds.

While anything is possible, it's not very likely that there is an internal trans issue causing this. Without looking at the wiring diagram, I couldn't say for sure how the circuit runs, but the switch would be in either the power or ground circuit. Either way, if you disconnect the connector from the switch, with the trans in reverse, you should have continuity between the two terminals of the switch. If you do not, take the switch out, and push the plunger in as far as it will go and check it again. If you now have continuity, take a good look at the switch you took out, and see if it looks like the end of the plunger is severely worn, or otherwise damaged. If it is, then, yeah, NOW you have an internal trans prollem.
If you DO have continuity with the switch installed, but no back-up lamps, then you have a wiring issue between the switch and the lamps, or no power/ground AT the lamps with the key on.
I'll take a peek at the wiring diagram tomorrow, and see what we can figure out here...

The switch checks OK, rang it out with a volt ohm meter.

Well then...that leaves only a worn or damaged detent on the shift rail...inside the trans....
So...if ya want the reverse lights to work, you'll either need to remove/repair the trans, or hook a switch up on the console or something. That sux.....