Sounds like serious alternator issues.
Load test the battery, but it sounds like an open or shorted or weak diode in the alternator bridge and/or a bad regulator. You can get a free alternator test at a Sears shop or buy a minitester (I have a Sun) for maybe $20 or so at AutoZone. The minitester may have three or four LED's that light up in a pattern that tells you if you have low voltage, high voltage, open or shorted diodes (essentially a high ac component in the rectified dc output).
OTOH, if you have an oscilloscope, go for it and look at the alternator voltage waveform. It should be a series of lumps, all the lumps on the same side (+) of the abscissa. Any lumps below indicate a shorted diode (replace the bridge or most likely the alternator if you can't get a bridge)...missing lumps indicate an open diode.
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