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The test for the alternator is to measure the voltage with the car running at the battery terminals-- you should see 13.7-14.2V, so your alternator is suspect. Seeing all the other dash lights light up is a classic low voltage symptom.
It seems that the 70 series also suffers from voltage drops on the B+ cable-- you should look into testing this as well-- there are good procedures outlined for measuring the voltage drops in the archives here.
My money is on the alternator/voltage regulator. When you get that sorted, take the battery to Autozone or someplace and have it load tested.
Good luck!
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