Bad batteries will cause the alternator to fail. Using the car as you describe can kill a battery and the alternator.
There are so many ways a battery can go bad, any battery 3-5 yrs. old is suspect, regardless how little you may have used it. Even the shock and vibration a battery experiences is detrimental and can kill it.
Plus, a battery has only about 5 - 7 discharges in it before it's done for. Each time a battery discharges, it loses it's ability to hold a charge, according to the SAE, by 15%, and the effect is cumulative.
I'm guessing you need an alternator and a starter (ignition) switch from what you describe. The new battery was not being replenished by the alternator. Chalk up 1 discharge.
These cars do have a constant energy drain from the alarm, radio, DME, etc., and this drain is significant, 2-3 weeks can drain the battery. Best thing to do is have a maintainer (not a charger) which you can plug into the lighter socket to maintain the battery health between use.
Cheers!
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