While sorting out the spaghetti behind the dash I took some ohm meter readings to verify continuity in my illumination circuit. I read 0 ohms to both ground on the black wires to the bulbs and also to the red 12v power. However when I turn them on they light properly and dim as they should. I replaced almost all with LED's but they all work.
Does anyone have an answer as to how ground reads to both sides of the circuit but it operates properly? (I took the reading with the headlight switch off)

Disconnect the bulbs (LEDs), then take the readings again - LEDs are diodes and you're measuring through them - unlike incandescent bulbs which would show a resistance, diodes would pass straight to ground.

Thanks Rusty - Is there a chance this is also the case with a breaker that trips? Although the breaker does not connect to the bulbs it does connect through the warning light panel on the dash and I can't get the gauges to register. They don't come on when I turn the ignition. Do I need some sort of resistor in series to provide a load for them?

This is only "in general", since my Breakaway is wired differently from Regencies (and most earlier Breakaways).

Anyway, the bulbs and gages are on separate circuits; if the bulbs are fed from the dash itself, it's still a separate branch circuit. It sounds like the bulbs are a separate circuit entirely, as the gages don't work.

Gages normally require the full 12VDC to operate, with the sender (which is the path to ground) in most cases providing resistance based on the temperature, etc. IOW, the gages (oil pressure, temp, etc) are voltmeters. The speedometer and tach need the full 12VDC to function as pulse counters.

In short, the problem is that the instruments aren't getting 12VDC power. There is virtually no chance that a bias/ballast resistor needs to be added (remember, the dimmer is a variable resistor, and is in that branch circuit with the bulbs).

It's possible that you knocked a wire off a terminal or didn't reconnect one after replacing the bulbs with LEDs.

A breaker is always (well, properly) wired on the "hot" (+) side of a circuit. A reading through the ground side of a circuit should show 0. It's likely to blow a fuse in the ohmmeter if resistance is measured across an energized hot wire. A reading from the hot side of a circuit to ground with the breaker open will show the resistance of the load in that circuit.