1991 Honda Prelude A few different problems

Okay, I just got this car two weeks ago. The people that had it said that the alternator went. So we charged the battery, drove it to my house, got the battery and the alternator tested by auto zone, no problems with either. So I got the spark plugs, oil changed, all the regular stuff. Except, I didnt get the radiator flushed right away. A few days back the battery light came on, but the battery was still on green and we asked around, but every said it must be a short in the wiring. So the day before yesterday it was raining at night, so I turned the defroster on. Mind you the fan had been running for a while every time we turned the car off. Well yesterday, I was driving for a while and the defroster was still on, so I believe that it made the car too hot which caused the fan to run constantly which killed the battery. The battery died and we parked. We got the radiator flushed while there thinking it was just the hot engine causing all these problems. But then when we turned the car back on, the battery light was flickering on and off. We went to auto zone thinking maybe the battery had just went from it being drained by the fan. But the tester said the battery was good. They charged it, and we drove again, but the light was still on. And today, it died again. We don't want to buy a new battery if this is not the problem. What could it be? Also, the car shakes anytime the speed is between about 58 and 70 mph. Before or after that though, it does not shake. What's up with that?