Engine stays running without key in ignition (and more!)

I'll get the battery on the charger now. I checked the regulator before I left town, however I checked its resistance, not the voltage it was seeing. I re-check and look for voltage this time.

Resistance check of a wire is good for checking for major failures.
Its not sensitive enough to measure the resistance that can cause voltage loss during some normal operating scenarios.

There's a little example in this post showing the effect of just .06 Ohms of resistance in a connection.
Short version is this:
V=I x R
Under normal operation, the alternator supplies maybe 5 amps to power the ignition and alternator.
V = 5 amps x .06 ohms
V = .3 Volts
Everything works like normal...

But lets say the battery is real low on charge and the resistance is in the alternator output wire.
5 amps to run the car and 20 amps to charge the battery all going through the .06 ohms resistance,
Volts reduced = 25 amps x .06 ohm
Volts reduced = 1.5 Volts

So in this scenario all circuits after that resistance location will see power at 1.5 Volts less than the alternator is producing at.