My 91 lincoln towncar stalls when warmed up.

I have had this trouble for some time. My car stalls when warmed up and eventually the battery dies. The jumper cables get really hot when I jump try to jump it. I don’t think the battery was over 1 year old. When I took the alternator in to the car parts place it was checked out. I even took it to another place just in case and it checked out ok there too. So I get the car jumped and it drives around the block (about 1 mile) and it stalls. I buy a battery charger/alternator diagnostic unit, hook it up and it is like trying to revive a coronary patient who has lived on twinkies, hot dogs and donuts for 20 years. No response the engine barely turns over. I installed a new battery and hooked up diagnostic unit to battery to monitor the alternator while the engine was running. At first the unit showed 100% alternator working, then as the car warmed up (~5 minutes) the engine started running rough and I noticed the unit display 0% alternator and the battery was draining very slightly. The engine continued to run rough and I gunned it. The engine stalled and I had a little trouble getting it to stay started. It kept stalling. Then after a couple of minutes, I started the engine and the unit displayed between 88 – 97% (but never 100%) alternator working again. I left it running and it seemed to be fine. It started raining so I had to quit before I could experiment any more. Any ideas? Bad alternator, even though I took it in already and it checked out ok? I thought the O2 sensors were acting up at one time. Also, could a vacuum leak somewhere be causing me a problem? Anything you give me is a great help. Thanks a bunch.