Hi neighbor!
The battery is a great idea and I'm not quite sure how it tested good. If it's a few or more years old it should go - especially if it's been run down as you said. They also take a VERY hard hit being dead when it's cold out which can kill a new one right away.
I'd seriously think about getting a new alternator. It may test out ok but they can be funky. Doable yourself if you have some tools available. Make sure belt stays tight too. Alternator and battery highly affect each other and either can take the other out. If battery was weak for a while alt probably overworked and they don't like that.
Do clean up or replace battery cable ends as needed. Remember to take neg off first and back on last! 5/16" or 8mm wrench or socket for those. If you can - check out the engine and starter motor end of the battery cables too. If this uses the remote spot to jump battery clean all those connections as well.
Good luck.
Ps: I'm about 20 min from intersection of 495 and "The Pike"
T
Tom Greenleaf - MetroWest Boston - USA
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http://www.autoacsystems.com/tomgreenleaf/