Well, there are a couple easy checks but you need a DMM (Digital Multi-meter).
The battery isn't exactly 12 volts. Each of the six cells produces somewhere in the neighborhood of 2.2 volts. If you checked the voltage between the battery posts (car OFF, battery cables disconnected) you should see somewhere between 13.2 - 13.5 volts. Since you just bought a new one, this is not likely an issue.
You can start the car and check the voltage between the battery terminals - this time, you should bget a reading of between 13.8 and 14.5 volts - this is your alternator voltage.
A bad battery can damage or even ruin an alternator. Your previous battery may have been failing long before you stored the car and caused the alternator to work harder which means that it ran hotter and/or was subject to voltage spikes as high as 45 volts which could easily damage the integral diodes which act as voltage stabilizers/regulators.
If it is the alternator, it's not difficult to swap it out. Before buying a new one, look into having your existing one rebuilt - there are lots of shops offering this at considerably lower prices than buying a new replacement. Good Luck
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