What sort of converter/charger do you have? Some had a sort of transfer switch relay arrangement for some of the DC circuits that would power them from the battery when the converter is off and directly from the converter output when the AC is on. (There's a separate lower current charger output on these units typically.) If that relay were bad or sticking or something it could cause the problems you're seeing.
I think some also had a manual toggle switch for the same purpose.
