Engine stalling

I left our lights on in our 99 Jeep Grand Cherokee for about 1/2 hour the other day and the battery (installed 8/07) went dead. Got jumped from a portable battery charger, drove about 200 yds. And the Jeep died. Got another jump from same portable charger and this time left the charger on for about 5 min before we started. Took it off and then let engine run at about 1500 RPM for about 15 min. Voltage showed 14, then drove to a car part store, thinking I was going to get a new battery. They hooked some kind of hand held tester to the battery and said it was OK and that our alternator was showing OK too. They suggested to drive it for about ½ hour before shutting off, which I did. It was actually about an hour before I turned it off.
I went to start it a couple of hours afterwards and it started OK (with very slight hesitation) but unless I kept my foot on the gas pedal, it stalled. That was Friday the 24th, today is Monday the 27th and the same thing is happening. It starts OK (with slight hesitation) and keeps stalling even after running it a couple of minutes as soon as I let my foot off the gas.
BTW-I should also mention that our battery cables are corroded and definitely need cleaned or replaced completely.
I’ve tried to troubleshoot online and from what I can see, my gut tells me it’s the battery and we need to replace it. Or could it possibly be as easy as cleaning/replacing the cables? At least we're hoping it's the battery and not the alternator:)