Charging system

If everything failed try a different good battery.
The reason I said this is because if you have a dead cell but the others are good you're looking at 10.5 volts but the car will start (sometimes) and it will go full field and you'll see that kind of voltage. If you don't have a load tester turn on the headlights for 60 seconds than check the voltage at the battery or go the parts store and have the check it. I have seen this before. 12.7 -12.8 is a good battery.