I put it into gear and it dies.

Specifically, 5.8v (and 5.2v at the coil) on the side with the brown and the blue on it. and 10.78v on the side with only the blue wire. and the test light lights up at both sides of the bulkhead and there is power going into the switch.
WHAT.
That's how the resistor works. When current flows through, voltage drops and heat is created.
With the key in run and the points closed in the distributor, current flows through the ballast. On the run side its system voltage, on the other side its something less. 10.8 Volts is kindof low. Maybe the battery is low but maybe not. Next time you test that, go quickly measure the battery at the same time. A tenth or two is no big deal. A full volt between the battery (or alternator) and the run wires indicates poor connections.
Ok so before when I was checking voltage at the ballast with that rigging above, the test light only lit up in the specific spot in crank. NOW, after only unplugging the connector before the switch and plugging it back in, the ballast has power.
Firewall connector or column connector? Regardless, clean them up and make sure the metal terminals inside aren't backing out.