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It most definitely sounds like an alternator that is not charging. A new or fully charged battery will allow you to operate the vehicle but not for a long amount of time. I would have your alternator tested. If you have a voltmeter you can check voltage output, with the vehicle running at 1500 rpm's you see about 14+ volts off the back of the alternator. Most Auto Parts stores like Auto Zone will check your alternator out at no charge.
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