1991 Toyota Celica alternator/battery issues

I replaced my alternator this June with a brand new from the manufacturer model. The car ran fine for a few months and then the battery went, so I replaced it as well(I had put a new stereo system in and a wire grounded out draining my battery or so I thought)The problem is now, a few months later, the battery light still comes on and it will pick and choose when to charge. I've been told that there was a faulty connection between the alternator and the battery, but I cannot find it. I've ran all the diognostics one can do without a computer and the output of the alternator is fine and the voltage of the battery is lower than it should be but not by much. There are no visable frayed wires or corrostions. This morning I started it and drove.5 miles to the gas station where the car ran for 10 minutes while I was getting my breakfast and as soon as I got back it and put it in reverse the radio cut out, followed by the lights and then it died completely. After a jump it started right up, ran fine, then I turned it off and it would not restart on its own, but started up with a charge no hesitation. After a charge I removed the negative cable and shut it off, and it would not restart but tried to. It still had lights and radio, though. I know the battery is not charging properly, and I do not want to by a new one every few months to keep the car going, nor do I want to replace the alternator again if it's only a connection. Any suggestions as to what I should do now?