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.
About 3 weeks ago my truck stalled out while driving.. would start back up for a second and then die.. parked on side of road with hazards on and they eventually died had battery and alternator tested at Advacnced Auto. They told me battery was bad (reading 11 volts) but alternator was good to go. Replaced the battery and it was driving fine until today when it did the exact same thing. The weird thing is it will start fine and then stall out after a minute and then battery light and check engine light will pop on but the battery gauge still shows it’s good. I’m at a loss here. Could it be the alternator?