Yes, I did check the meter read 0 when touching leads together. 

I do have a relay kit ready to install, but I was planning to do that AFTER I find the problem, so as not to introduce a new variable.

However, I'm now thinking that maybe the problem is the headlights are just drawing too much current for the switch, so maybe I should do the relay now.

It's a sealed beam 45/60. Is that modern?

To add to what ve9aa said (Thanks Mike - good advice), the multi-meter has an internal battery that is used for measuring resistance - it has to send an electrical current for the test. If the battery is very low you won't get an accurate reading. There may/should be a "Zero" setting knob to correct the needle reading when the probes are touched together.  (I don't know beans about digital meters though!)

To know if it really was 8 Ohms or not, did you touch the 2 meter leads together and see what the meter read (before making ANY measurements)?
Hopefully it was less than 1 Ohm.  If so, then go ahead and make your readings on your lighting circuit after that.

I would think a short chunk of wire inside an engine bay should be well south of 1 Ohm as well.

8 Ohms seems high.

Where there is any resistance you'll have heating. 

YMMV

modern halogen headlights should be run on a relay, this protects your switch and wiring from the power they draw. You can find relay kits on our site, but if you're handy with wiring you can find how-to on youtube maybe and get a standard relay at your local parts store for about $10 maybe. 

Yeah, I think that Moss Motors sells a headlight relay kit that will save your switch from overheating.

Hi!
I've heard several say to run some headlight specific relay kit so that They take the power instead of your switch.
Best to you and yours,
MSH

No idea what the resistance for headlight wires should be but the two should be the same as they would deliver full voltage to the low and high beam filaments. The beam of light changes because of the position of the filament in the reflector and the wattage of the filaments themselves (the high beams may be brighter).

The same would apply to double filament (e.g. 1157 bulbs) rear brake/tail or front turn/marker lights. They all should get full voltage and it is the filaments that change the brightness.

To check for grounding shorts:
With all switches off, test each wire from its terminal to a good bare ground. Pull the terminals off the lamp fixture and put the meter probe in each one. You should get infinite resistance, i.e. no connection. Black wires should be ground wires - they should peg the meter needle at zero resistance. 

1976 Pickup. Wiring diagrams don't seem to match the colors I have so working pretty blind.

Headlights, flashers and brake lights failed. Replaced switch. Headlights worked for about 10 seconds and then failed again.

Assuming short. Will install replay once I have the issue identified and fixed.
Be patient with me and make sure I have covered the basics please Last time I dealt with electricity was back in A level physics.