help trying to make this thing run right

[53farm]

I've got a '53 3600 with a 235 in it. I have replaced the plugs, wires, cap, points, and condenser. I have verified that all fuel lines are clear, albeit old. I have a new gas tank on the way for whatever thats worth. It has been converted to 12 volt by previous owner.

When I bought it, the guy said he had been keeping the battery disconnected because something kept draining it. It did run when I bought it, and I drove it into my driveway. I've had it running/idling on its own (once off a gas can and once off its own old gas tank) on two occasions in the last couple weeks. When I put it in gear to back it out of the driveway, it cuts off after a foot or two. It has done this on both occasions. After that it acts like the battery is dead, or almost dear as it cranks really slow, and I have to use the starter/charger to start it again. Once I have it started again, I have to keep giving it gas, otherwise it would cut off.. This was always on the second start however. It would run on its own, without me giving it gas, on the first start/run before I try to move it.

I know the battery that I'm using was good to start with because I took it out of my running/regularly driven '63 Bel Air. At this point I'm pretty sure this is an electrical problem, no? Do you all think this is a wiring issue, or a bad alternator? I have a new, but basic, 12V wiring harness to put in it, but was trying to wait until I could get some more paint/metal work done. I really need to get this thing running regularly so I can at least be able to move it around before I continue progressing on it. Thoughts or advice is appreciated. Thanks!