YES, your electrical system is split, AC voltage which is 110 and DC voltage which is 12 volts DC, when you are plugged in the AC items would be the outlets, microwave etc. The DC items would be lights, water pump, fridge, AC unit, water heater etc. Some of the 110 items require 12 volts to turn on such as the fridge and AC unit.
SO I suspect all the outlets are working but you are missing ALL the DC power, if the battery is not getting charged when the power goes out then you will not have that DC voltage.
The converter will take the AC voltage and charge the battery..
IF you purchase new batteries it should take care of the problem however you need to verify that converter is working, when you plug in you should see the battery voltage rise, as long as they are charging then you are set....
Hope this makes sense, if you have more questions or concerns please feel free to contact me, IF you are satisfied with my efforts please feel free to select a rating, it is the only way I get credit for helping.... Robert