I know this thread is ancient, but If you ever come across this problem on a 7.3, it is very possible that your IDM(Injector Drive Module) is about to fail. May have water inside causing this symptom. I changed my IDM, and lo and behold, it went away.

A simple $3 digital meter in the cigar lighter would see a failing alternator that can and will take the batteries with it, quickly. The thing to watch is the morning voltage and drop under load, on the first start.