Battery dies after driving

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Hello! My '76 is acting up. It starts no problem when I take it out. If I drive it to work (half an hour commute) it's fine. But if I shut it off and try to start it again immediately the battery is pretty much dead. I can let it sit for 15 minutes and suddenly the battery has enough life to start it. Battery is brand new, and it replaced another new battery that I thought might be bad. I had the alternator tested (it's a remanufactured 12si) and they said it checks out fine. Engine is a 350. Any thoughts on what could be dragging it down? Should I get a new alternator? Thanks for any help.