I hate to tell you, but it sounds like your alternator. I replaced four batteries in my 99 Suburban before I took the alternator off and had them do a proper bench test on it!! And guess what, it was bad all the way around!!
Oh, and about testing for the shorts? I was told the same thing, and then I did that on my 2003 Ford Ranger, I did that on the neighbors 1999 Chevrolet Suburban, and on the 1999 Contour, they all lit up, as though they had a short. After talking to more people, the explained that there is a lot more to checking for a short!!
I'd have the alternator properly tested. It killed three batteries in one year for me!!
