Sounds like converter is not working properly. You should use a voltmeter and check to see what the output is.
You still need to use battery for all of lights and other things that use 12V electricity. Your converter, being on shore power should keep battery charged.
When battery has full charge and not connected to shore power do the 12V items work? If not, that could be a battery disconnect switch.
2016 Ford F250 XLT 4X4 6.2L
2015 Rockwood Signature Ultralite 8329SS
Nights camping in 2016: 19 nights
Miles traveled in 2016: 2,575