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It does sound like the alternator. The best way to test it is to have it bench tested. Places like Auto Zone will do this. If you have a volt meter you can check voltage at the back of the alternator on the positive cable. With the engine running around 1500 rpms you should have 13.5+ volts. I would also have your battery load tested if it has been drained down numerous times. Sounds like a diode in the alternator is bad.
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