95 Blazer and 96 Cavalier Battery or Electrical problems?

This just happened with my Cavalier the other day. I had the Blazer sitting for a good year or so because the front drive shaft came loose and punched a hole through the transfer case. It's an all-wheel drive. I finally got some money up to get it fixed, but it ended up costing me more than I thought it would (3600.00) to fix the shaft and transmission. So it sat at the shop for four months. Got it back, sounded like crap. Figured it just needed driving. Besides the fact that my new tranny has a crack in the transfer case and was leaking fluid, smelled strong gas smell too and had fuel leaking from fuel pump which eventually stopped. The problem is, they told me I might need a new starter because it seemed to hesitate on starting. It did, but while driving, the battery warning light came on and then the check engine light came on, both solid. I looked in the manual and it says to turn off A/C and/or radio if that happens, which I did. Then the check engine light goes off. But I notice on the Blazer the voltage of the battery is down to almost orange at like 9 or something, so I stop to have the battery tested. It's "good" and "fully charged", so they won't charge it. So, it starts to rain, I turn on the headlights and the windshield wipers and the voltage drops down into the red zone. I just make it home and turn everything off and it won't turn over now. Get it jumped and it stays on for a minute or two, but then the light comes on again and it just dies. Same thing with the Cavalier. I stopped at a store, came out and it wouldn't turn over. Tow guy comes, jumps it, but the instrument panel isn't on, and then it shuts off. He comes back, jumps it again, tests the battery and says it's fully charged. We drive off, then the thing shuts off on us while we are driving! So it had to be towed home. Any ideas before I take either one of these into a shop so I know what to expect? Friend of mine said it could be alternator, or could just need spark plugs changed and tune up. Any help is appreciated!!

Rule no 1
Make sure the battery and cables and terminals are good.
Have you got a volt/ohm meter to check it?
MT