suggest you Google: "The 12 Volt Side of Life I & II". you'll learn a lot about the electrical system in your trailer.
the short answer is yes. you have somethings that are only 110vAC and somethings that are 12vDC.
the a/c(if you have it), the microwave, the 110v outlets only work on shore power or a generator.
somethings can run on both, like the fridge.
the lights, furnace, roof fan, electronics for fridge/water heater DSI run on the battery.
the converter will simply recharge the battery to run these things when on shore power.
if you have a bad battery, your converter will not be able to keep the battery charged, therefore your 12v things will stop working.
and Zoe the Wonder Dog(R.I.P.)
'07 Roo 23SS pushing an '07 Chevy Avalanche
Equalizer WDH and Prodigy BC