I would suggest having your converter checked, as I suspect it is probably bad. If it is bad, then replacing the unit would be your recourse. Your 120 vac as well as your 12vdc loads are all distributed through the RV's converter. Even the power to your 120vac outlets comes through the converter first. They are not powered directly from the shore power outlet to the on-board 120 volt outlets, but are powered through
the converter. If you are blowing the main 20 amp fuses, it's telling me that your converter is most probably shorted out internally. This is not an uncommon problem with converters, especially older ones, but brand new ones have also been known to crap out shortly after being installed.
I would also have the battery tested again...they do go bad, and when they do, it's usually "right now".
Your trailer lights (i.e. tail lights
, brake lights, turn signals,) and your alternator's battery charging output to the trailer battery (while towing) derive all their power from the towing vehicle while you are underway and electrically connected to the truck (I say truck, but whatever vehicle you tow with). First, I would suggest having the towing vehicle's plug-in outlet checked, (if you aren't comfortable doing this yourself, any hitch shop can do this usually for a nominal fee,) to make sure all the power and indicating circuits are functioning properly to this point. If your truck's plug checks out ok, then I would suspect the problem to be in your trailer's wiring, usually caused by poor grounding. This would also contribute to a whole lot of 12 vdc power problems. These would be good things to check first, as they are generally the causes of many power problems with rv's.