I bought a new battery for my 96 Explorer 4WD V6 5 days ago. I stopped to
get gas, and had to get it jump started. I went back to AutoZone, they
checked the battery and the alternator. They said both were fine. They
recleaned the terminals and it started up. I also just had the transmission
rebuilt about 3-4 weeks ago.

Today, my it wouldn't start. I had to jump start it, I drove it for about
5-6 miles afterward and then it still wouldn't start. So I had to jump start
it again. On the second drive, I noticed that if I turned on the A\C with
the blower on high, the engine would cut out for just a second, almost to
shutting off. Also the battery light would come on in the first few minutes
when I hit the brakes. After 1/2 hour drive, the engine won't even crank.
While driving I watched the voltmeter, it was OK, it bounced slightly when I
would turn on the turn signal, it's always done this. And would also drop
slightly when I rolled a window up or down, again always done this.

Did I get a bad battery? Or is it possible something else is at fault. This
only started after getting the new battery.

Thanks.