Will bad house batteries cause 120v shoreline power issues?

Get yourself an PI EMS, like a PT50X 50A model and see what kind of AC power you're getting from the post. It'll help eliminate park power as being involved. Or, just try another post or outlet?
It's still probably old/weak/failing batteries, but you never know, and there are many stories on here and elsewhere about bad park power causing all sorts of problems.