Back up lights on a 68'

Hmmm, need to keep things straight. If you have 12 volts power in one side of the plug from the firewall, but no lights, the problem is between the trunk, and the plug.
Make sure you understand the circuit you are working on. When you seperate the plug at the firewall, one side goes down to the switch on the trans. The other side is the power and the lights. Unplugging the connector simply removes the neutral saftey switch from the circuit. Moving the shifter with the plug disconnected will have no effect at all. If I understand what you wrote, if you proved that you have 12 volts, and you shorted the two terminals together, the black with red strip wires, and no lights, then the neutral safety switch is no longer a concern. You can test that later once you get the backup lights working.

When you seperate the connector, one side goes into the firewall, the other end goes down to the transmission to the switch. Pull the connector apart. Set the one end aside that runs down to the trans. Take the other conector in your hand and identify the two black with red stripe wires. Short these two wires together and see if the back up lights come on. You may have done it that way, but from what you wrote, it wasn't clear. Let us know what you get with that test first.

IF, and only if, the lights come on, the problem lies with the switch on the trans.

If they do NOT come on, verify again that one of the two wires has 12 volts (good test light indication) on it. Again, make sure you have the ign switch in the ON position or the ACC position.

If you have a good test ligt indication, leave the wires jumpered. Go to the back up lights and make SURE the bulbs are good. Pull the bulbs and CAREFULLY probe the center contact and see if you are getting 12 volts to the center contact in the bulb socket. You can tape up the probe to make sure you don't short out the circuit and blow the fuse.

If you have a good test light indication, check the bulb. If not. Write back, there are two other places that the backup lights can be disconnected. We can get to that later.

It would be odd that you have two problems...it would mean that the switch is bad. You should have continuity on the Black w/red strip wires in reverse, and continuity on the red w/black stripe wires in park and neutral. If the car starts and runs with the plug conncted, then the neutral part of the switch is good, and your test is not being done right. Actually, it's just the start circuit-ign sw, connector, neutral safety sw, solinoid. So if it cranks at all connected, then check how you are doing the test.
If no lights with the connector jumpered, and the ign switch is ON or ACC, there are a couple conections. THere should be two pigtail connectors for each backup light in the trunk. One is 12 volts from the connector at the firewall, and a pigtail connecter for the ground in the trunk. Look at the taillights. THere should be a harness dropping off to the bottom of the qtr pnl. Check these pigtails and make sure the ones that are tied to ground are good and tight and clean.
There is another connector at the drivers side kick panel. If you remove the kick panel and contort yourself, you will see the trunk harness coming out at the top of the kick panel. THere are several conectors. One should amazingly enough be a black w/red strip wire. Make sure this connector is tight.
If you do have 12 volts at the conector on the firewall, and no 12 volts at the bulb sockets, or at one of the pigtails in the trunk, you can split it here and check for 12 volts. Let us know what you find from there.

Alrighty then! Since you're feeling froggy...you "can" take the neutral safety switch apart. Drill the rivits (not the plastic) and carefully seperate the halves of the switch. It's probably too far gone, but I have fixed one...someone pulled so hard on the harness that they pulled the switch apart enough for one of the contact strips to drop loose inside. Didn't break the plastic. I put some small machine screws and nuts back in place of the rivits with some locktite. THat was a 68 coupe too...it could also be a wire corroded off the contact. Can be soldered back on. Switches aren't too expensive either. Well done though!

Yea, I know. THat one was an easy fix though. Usually (99% of the time) they are toast. Defintely get a new one. Will be more reliable down the road. Now you can stop backing into trees cause you couldn't see them!

When you replace the switch remember which way the kick down lever came off.
Your Gramps sounds like my older brother