Bad alternator or bad battery?! Help?

I have a 1998 Honda civic, a few weeks ago I replaced the battery in it, it was over 10 years old and always needed to be jumped. Okay so two weeks ago I replaced the alternator in it. It was bad also. Car drove great. Two days ago the battery light came on, but I just ignored it. (I know stupid) Car still drove ok though. Yesterday my gages were jumping everywhere but I still made it to where I needed to go. TODAY I started it, lights were super dim, but I made it about 5 miles down the road and it died. Jump started it, drove it about a mile, slowly lost power, gages jumping everywhere and eventually steering locked up and car died again. Tested the battery it tested at 11 Volts so I left it on the side road. I am a woman, not car savy and dont want to get ripped off, I have bought an alternator before for another car and it did this, went bad, gages were jumping everywhere and car would barely shift. Replaced the alternator, problem solved. That one however never left me on the side of the road. Any pointers would be great. Thanks!

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