1975 Dart Sport - Alternator Issue?

If it only dies while slowing down, that sounds like a carburetor problem. Does it idle properly otherwise?

It does seem to idle fine otherwise. There are times however that I have hooked up a multimeter and ran the car with the accessories on and the battery starts to drain - to the point where it starts to rough run and then dies.

I have a feeling that I've got multiple issues going on here.