I'm really stumped right now as I haven't had the lights in my gauge cluster or my tail lights for 2 weeks now. I replaced the headlight switch, replaced the melted connector for it, and even checked all the fuses for any dead ones and still nothing. My car is an 84 Camaro, so I don't have any fancy relays on the computer. What grounds should I check if that's the issue?

 

Not a "ground" problem.

Replace the TAIL fuse. (not "check"... REPLACE)

 

Are you saying that Both the 5A "INST" Fuse and the 20A "TAIL" Fuse have 12 Volts at both Probe-Points on each Fuse?

...But still have to Gauge Illumination or Parking-Lamp Illumination?

Are you sure you had 12 Volts at the RED Wire that Connects to the Head-Lamp Switch?
This (Going down to the Starter Motor Fusible-Link A) Power Source (which does splice out) is responsible for power at the Two Fuses that I mentioned and their Circuits pass through the Headlamp Switch.

The Brown Wire is the Parking-Lamp Circuit Power to those associated Bulbs.

The Green Wire is the Power to the Fuse for the Gauge Illumination.
Gauge Illumination Circuit Wires are Grey.

 

Did that and still nothing. I also have a new headlight switch too and that didn't help

 

OK then, you'll have to do some troubleshooting.

What's the voltage at the orange wire going into the headlight switch? Then, if that has 12V, at the brown wire coming out, with the switch in Park?