Doesn't sound like the battery to me
I don't think, even if the alternator/charging system was dead, that it would prevent you from running the car for short periods, or at this point, keep it from geting it started. If is it turning over OK (If is sounds "Normal") should be enough to start and run it. (Especailly since you re-charge it and it is at 13v.)
You may have a battery cable (Short) or other wiring problem that is killing the battery.
I just had an alternator die on my car, but it ran fine. I was driving around town, headlights on, on a 3 year old battery. It was a little less responsive than normal, and I kicked the scanmaster over to read volts... 10.9. Obviously, I headed home for diagnostic work, but the point is, it didn't die. I suppose the alt could have an internal short, but you could take the lead off temporarily and check that.
I would look for melted insulation on the positive battery cable under the down-pipe, loose connections at the starter since it serves is a main power distribution point. Look to see if all the ring terminals are still connected there, and also check the fuesable links near the starter. Check the ground points to. The starter has a high current rating and may still crank even when poor/high resistance grounds are present.
If all these basic items check out, check the fuses. A blown fuse may lead you to the source of the problem as well.
Check to see that while the key is in the run position (Before cranking) that the Check Engine Light is lit. If not, let me know. I copied someone's expert troubleshooting list if it isn't lighting up.
In addition, if all the above is working, you need to see if you have both fuel and spark being delivered while cranking. I have another list of items to check these items as well.
Bottom line is, even if the alternator is disconnected, and your battery is fully charged, it should be enough to start and idle the car for at least several minutes. If you doubt the battery, replace it and get on with the troubleshooting!