First of all, it is not an inverter that you are having trouble with, it is a converter. This detail is not nit-picking, the two devices are very different, do different things, and unless you name the right part when you go buy one, you are guaranteed to get something that will be of no help solving your problem. Just sayin' . . . .
Secondly, unless you have a relatively new (meaning a couple years) 3 stage converter, leaving your rig plugged into shore power continuously is guaranteed to kill your battery in fairly short order. Given the symptoms you describe, it sounds like you need both a new CONVERTER and a new battery. Spring for the extra bucks and get one of the 3 stage units. It's worth it.
Finally, to check and see if indeed the battery is dead, take it out and go to a place like Pep Boys, Advance Auto, Auto Zone, or one of the battery shops. They can charge it up and test it for you, and will do it for free. If they tell you the battery is dead, buy a new one from 'em, but don't install it until you get the CONVERTER replaced.
(NOTE: be sure to mark the wires you take OFF the battery so you know which goes to positive and negative terminals. You don't want to get 'em on backwards lest your problems really take off.)