"Shouldn't the cars electrical system run completely off the alternator and not use any battery power while it is running?"

Not really. The alternator generates energy, but it also has to work WITH the battery under heavy loads because it can't supply enough on its own. Air conditioning is a heavy power draw. Could your alternator also be flaking? Sure. But by the sounds, I'd say most likely it's just a weak battery not being able to live up to its end of the power bargain in your system. It recharged the battery when you were not drawing extra power, so that's a good sign. You'll know pretty quickly once you replace that battery.