Hi, Crankin; First, 11V in the battery is just telling you that it needs to be charged. You cannot test a battery that is below 12.4V. Soooooo, we're back to the alternator theory. Again, you cannot accurately test an alternator with a low battery. Nothing against Advanced (they are our main parts supplier), but they are not auto. techs. It really sounds to me that you are running off of battery voltage, rather than the alternator, until the battery get so low that the computer takes a dump and the engine dies. My suggestion would be the have the battery fully charged, then take the vehicle to a reputable shop and have the alternator tested.
Good luck.