Backup lights not working

Your thinking got derailed because the neutral safety switch and the back-up light switch are two totally different switches but in the same assembly. The first diagram shows the back-up lights are on the light green wire circuit. The second diagram shows the connector for the switch assembly. The third photo shows what the switch looks like. Check for 12 volts on the blue wire by back-probing next to it while it's still plugged into the switch. If those 12 volts are missing, we'll have to go a different way, as into the fuse box. I'll try to figure out which fuse to look at.

If you do have the 12 volts on the blue wire, check on the light green wire when you shift into reverse. If 12 volts doesn't show up on that wire, the switch is defective. You can verify that by connecting the blue and light green wires together with a small jumper wire. That will turn on the back-up lights.

If you do get 12 volts on the light green wire, we will have to work our way toward the rear to find a break in that wire.

These articles may be of help to others researching this topic:

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

https://www.2carpros.com/articles/how-to-use-a-voltmeter

Dandy. It's always possible to have two different problems, one for each side, but for now let's assume just one defect is causing both lights to not work. To be safe, you might want to check for 12 volts to the terminal in one of the back-up light sockets corresponding to the light green wire. If you find 0 volts when the ignition switch is on and the shifter is in "reverse", there has to be a break in that light green wire. This harness is going to run to the rear along the frame rail. The connector I have an arrow pointing to will be in that harness and will make a good test point. Unfortunately, they don't show where it is. I have a suspicion it's going to be close to the rear of the truck where they could plug in different harnesses on the assembly line for different models, or it would be a nice place to plug in an adapter harness for a trailer connector. That connector is also a good place to find corroded terminals.

The best is to back-probe the wire at the connector while it's still plugged in. Second-best choice is to use a test light, especially if you have to disconnect the plug. If you use a digital voltmeter when the connector is unplugged, corrosion around a break in the wire can pass enough current for the voltmeter to incorrectly pick up and show 12 volts when you can't get enough current through that corrosion to run two bulbs. That's why test lights work better for this type of test.

Check for 12 volts on the front of the connector, then, if you have it there, check on the other side of the connector. If you do that first before unplugging it, that can show if the break is inside those terminals before you disturb them.

To be technically correct, that would be an "open" circuit, meaning it has a break in it. A short would allow too much current to flow and would blow the fuse. Terminology aside, it sounds like you've located the area of the defect. The best approach now, if possible, is to shift to "reverse" with the ignition switch on, get it to the defective state, meaning the back-up lights are not lit, then test at each point in the circuit, in order, until you find where the 12 volts is lost.

Start at the dark blue wire. Back-probe that wire in the connector on the switch. Assuming you find 12 volts there, next, try to reach under the connector and touch the test light's probe to the corresponding terminal on the switch. If 12 volts is missing there, that terminal is spread or corroded and making intermittent contact.

If you do have 12 volts on the terminal, check on the other one corresponding to the light green wire. If that's the first place you find 12 volts missing, the contacts inside the switch are intermittent. Switch contacts are rarely affected by wiggling the connector. If doing that makes the lights flicker on and off, it's almost a sure thing one of those terminals is loose or corroded.

One more thing to be aware of. Your truck was built at a time when GM was using aluminum wire on some models instead of copper. Any place two different metals come in contact with an acid, you get "galvanic" action, meaning corrosion. There were many problems with the aluminum wires riveted to steel strips in the fuse box with brass rivets. Add road salt from your feet and it was fairly common to find loose / corroded rivets in the fuse box when it was in front of the brake pedal. You'll see that by the lights flickering on when you wiggle the wiring harness going into the fuse box. If it's fairly dark under the dash, you may see small sparks occurring at the point of the intermittent connection. You can't find a better clue than that as to the location of the bad connection.

If you do have aluminum wire in your truck, the insulation will be a translucent color, not a solid color like we're used to seeing. I never allowed my students to pierce a wire's insulation to take a reading, but that's even more important with aluminum wire. With a test hole pierced through the insulation of a copper wire, that copper will corrode within a few years, and that corrosion makes it impossible to solder to. With a hole in the insulation for an aluminum wire, that wire will corrode into aluminum oxide within a few weeks to perhaps as long as a couple of months, then you'll have another intermittent problem to diagnose. You'll see through the insulation the wire has turned to powder for a couple of inches from the hole. Aluminum oxide is also an insulator that prevents further corrosion, however, in areas where vibration or flexing is an issue, that knocks off the oxide coating allowing further corrosion to occur.

Let me know what you find.