I would say you converter is going bad.
You could try checking the battery. How old is it?
Do you have a volt meter to check battery voltage and voltage output from converter?
Yes, it is a good idea to go through and check all 120 volt connections in the 12 volt DC distribution panel. Disconnect from shore power first. Check neutrals (white wires) as well as the black wire coming from shore power. I always suggest disconnecting the wires, inspect they for dark burnt areas, coat with electrical de-oxidation grease, then reconnect. Do on at a time so as not to make any errors when reconnecting.
Can you see what make and model you have?