Take a volt meter and measure from the panel that's alarming on low voltage and confirm there is a low voltage. Then measure the same spot with shore power attached. If you truly do have low voltage at the panel when shore power is disconnected, then trace that wire back to the battery, measuring along the way. Be sure to use the same ground on the meter as you trace the hot, positive lead.
If that checks, then start checking the ground wire circuit from the panel to the battery negative lead.
I suspect you have a faulty ground between your battery and the panel, making the panel think you have low voltage.