I suggest you disconnect shore power and then read your six volt battery voltage. If it is less than 12 volts you have a bad battery or batteries...but this could be caused by a cell with low water. I would use a hydrometer and check each cell for proper charge..very cheap tool, simple to use and can save you a lot of time. A bad battery will cause your converter to work harder than it should. It cannot charge a battery that is no longer capable of holding one.
Checking lead acid battery with hydrometer