Think of your batteries as a buffer. Even on shore power they will store power and if your converter doesn't have enough juice it will supply that amperage to help out. When the amps fall off the converter takes back over and the batteries top back off for the next time. If your batteries are going out they can't supply the needed amperage to the system. That makes your converter work extra hard and your voltage will dip causing the lights to dim and maybe even flicker.
I vote batteries are toast
Chris