Harbor Freight has a Battery and Alternator testing tool for just a few bucks, well worth it. Yes, your bad alternator may have damaged your battery. My battery died in my Jeep and I looked at the alternator. It turned out to just be a corroded connector at the alternator not allowing the battery to charge, a simple cleaning and driving a few times and all is well. Connections being metal, it is easy for corrosion to foul up the works so I always check these first! It's my opinion that this might be a solution to many of our CPS problems.