Other things to consider:

1. LED bulbs would put less strain on the circuit.
2. Check the side markers for corrosion. The front ones especially are subject to water infiltration and corrosion.
3. Rebuild the headlight switch and clean the contacts.

Head light relay?

38 minutes ago, Patcon said:

Head light relay?

I haven't finished designing that. I have a plug & play in my mind's eye, but I need an intact 78 or 77 to study and test with.

59 minutes ago, SteveJ said:

Other things to consider:

1. LED bulbs would put less strain on the circuit.
2. Check the side markers for corrosion. The front ones especially are subject to water infiltration and corrosion.
3. Rebuild the headlight switch and clean the contacts.

I'll have to check these things soon. The connection was hot enough to burn my skin when I checked it this morning after driving it. So clearly the problem still exist. @Captain Obvious I'll bring a lot of ice cubes with for my trip to Houston, TX this weekend 🙂 .

Yesterday when I was trouble shooting I found there was no continuity with one of the license plate lamps so I fixed that. I'd bet there is similar corrosion issues around a lot of the front lamps.

20 minutes ago, dylonwalk said:

The connection was hot enough to burn my skin when I checked it this morning after driving it.

Yeah, that's not good. Heat comes from resistance combined with current. If you've confirmed that the current is normal, then you're left with resistance.

Have you opened up that switch and had a peek inside? There are plenty of threads on the subject:

 

@Captain Obvious I haven't taken it fully apart yet. It turns out the dash lights still don't work. I'm sure that's playing a part in the resistance issue. So I'll need to fix that before I open up the switch.

Edited November 12, 20213 yr by dylonwalk

11 minutes ago, dylonwalk said:

@Captain Obvious I haven't taken it fully apart yet. It turns out the dash lights still don't work. I'm sure that's playing a part in the resistance issue. So I'll need to fix that before I open up the switch.

Did you adjust your rheostat for the dash lights to confirm?

Great question!  Thanks Steve! 

So I fixed me issue. I rebuilt my headlight/combination switch just like they said in the forum linked below. The only thing I didn't follow was unsolder the 4 wires on top of the board.

https://www.zcar.com/threads/headlight-combination-switch-repair.311982/

The switch no longer gets hot. Thank you to everyone who helped me!

Edited December 13, 20213 yr by dylonwalk

I did my '77 too. Made it like new I suppose. Flipping those plates is the money shot.

I followed this...

https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.kamikazeracing.org/dl/ZTech/Rebuild_240Z_combo_switch.pdf&ved=2ahUKEwinvMeuz-H0AhX0SDABHZA6DqsQFnoECA4QAQ&usg=AOvVaw1MgWNgBd3yM5WQ0jnGnjgR