honda civic 96 dying battery, broken speedometer, please help!?

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I have a 1996 Honda Civic, yesterday the speedometer/odometer stopped working, it just stays on zero. Then later that night, i was driving home from work and my radio stop working, my lights went very dim then it died altogether, after I jumped it, it died again after about 5 mins, took 3 jumps to get home. This morning I took the battery to Walmart to have it tested, they said it was bad even though it was only about 3 years old, so we bought a new one. After installing it, it started up perfectly, the check engine light (which came on a little over a month ago) went off, but after moving my car to another parking spot it came back on. Also, My power windows which have not worked for about 5 months or so, started working again, but stopped working soon after. My power locks also sporadically stop working. I decided to go to Autozone to get the alternator checked out, they said it was weak and that i need a new one but the one i have is less than a year old. sounds like an electrical problem? I dont want to spend the money on a new alternater if i dont have to!