[65comet]

My first guess is the dead battery was caused by alternator troubles. Take it to a auto parts store and have them check that it is putting out enough voltage. If it doesn't put out a constant high enough voltage the computer will do all kinds of strange things.
If it is fine, then my next guess is a bad relay. If relays check out, next guess is the ground wire from the battery to where it connects to the block may be loose or coated in oil/grease/dirt and lowering the voltage flow. After that, I'd scratch my head some and start checking wires and connections. Oh, and the most obvious is to connect an OBD-II and see if there are any codes, auto parts dealer can do that also. Good luck.