No Will, the 12.53 was taken at the battery itself. The engine was off and the positive was disconnected for reading voltage potential.Do I understand you correctly? Do you have 12.53 volts with the engine running. If so, do yourself a favor and shut it down and put the battery on a charger for a few hours. An alternator is not a battery charger and my guess is it will be hot enough to cook your dinner on in no time.
As for the other, if you follow exactly what I said and you get accurate readings on your current draw, I can tell you what to do next. When we figure this out we can look at your other problems but for now, lets get current draw solved. Stay on course. Pulling bulbs and other things are guesses. I do this for a living.