1997 Cadillac gradually loses battery charge, along with some other minor issues that may be related. Any tips on how I might tackle this?

Hello everybody! A while ago, my 1997 Cadillac Seville started accumulating minor malfunctions before eventually becoming inoperable due to the battery being unable to keep its charge. It started off with the lights on the dashboard intermittently going dark, and the twilight sensor stopped working altogether, forcing me to turn on the  headlights manually. After that, the heater blower went out with an audible pop and a burning smell in the cab. Some time after that, the battery stopped holding a charge on its own. I don't believe it's the charging system, because the alternator is charging the battery. When it's being driven daily, it'll hold enough charge to start and run the car for several days before it goes dead. When it's not being driven, it rarely makes it more than 24 hours. When it goes dead, the spark plugs stop firing and the lights dim significantly. Several friends and family members who are experienced amateur mechanics seem to believe that I have a short somewhere that is draining my battery, as well as causing the issues with my dashboard lights. The battery is a newly bought refurbished battery. Does anyone have any other tips or theories as to what it could be or how I could fix it?