Your measured voltage indicates battery has lost quite a bit of charge and needs to be fully recharged by plugging into shore power or using a portable battery charger. When plugged into shore power use your meter to measure converter output to ensure it is putting out at least 13.6 volts, higher would be better indicating a multiphase charging circuit is contained within.
Your battery does take over the duty of delivering power to your DC devices such as your lights, furnace, water pump, and fridge control board which you will run in propane mode. Microwave, AC, and all household type outlets require AC voltage and will not function once unplugged from shore power unless a generator or inverter is utilized.
A battery cut-off switch can be helpful if you need to keep battery from discharging when not camping, we have one installed from factory but rarely use it since we store our rig in our backyard plugged in 24/7. Before starting out ensure battery is fully charged and verify your tow vehicle is providing a charge to your battery while driving down the road.
Running furnace excessively and using lots of lights can quickly draw down a single battery, hopefully your newer rig came with LED lights which would alleviate that issue. If your fridge has the option to reduce condensation make sure it is turned off, many times they are unknowingly activated. A single battery can be depleted quickly but can last a couple of days if one is frugal with DC power demands.
