1991 Toyota Camry, Mileage 225,000

Hey guys. I have a 1991 Toyota Camry with over 225,000 miles. The car is in really good shape. I have put a new motor and transmission in it. Anyway. Recently I was driving and the vehicle just died on me. I was able to get a jump and then about 1/2 mile down the road it died on me again. The first thing I did was remove the alternator and took it to one of the Toyota dealers. The Alt. Was bad. So I purchased a new one from the dealer, so I know that it is a quality unit. After installing it, I noticed the battery was bulging on one side of it. Luckily enough I had a spare battery that I had purchased about a year ago. I put it on the charger for about 2 hours. I put it in the vehicle but to no avail. I jumped it and it ran fine. While driving it I noticed that the lights were a little dim. If I rolled down the window, that made the dash lights go even dimmer. Just about anything I pressed, it did the same thing. If I shut off the vehicle it will not start, I have to jump it. What else am I missing. Does it sound like my battery is junk? Does it sound like maybe I need to invest in a new wiring harness? Please help. I am running out of ideas.