WHOA: What I wrote below is true, and may affect somebody else, but your question is how to run 120-volt AC appliances from battery when 1. Genny isn't running and 2. You aren't on Shore Tie 120-VAC? YOU CAN'T. Unless you install an Inverter to take 12VDC from Battery (Chassis OR Coach) and use it to power appliances through extension cord/power strip --OR-- modified coach wiring. And, AC loads translate to a LOT of Amps when you try to make the AC from a 12-Volt DC battery source. You can buy say a 400W (at 120VAC) Inverter and do things like watch TV. Many inverter systems won't run A/C, Microwave, etc. That takes a massive inverter and lots of battery capacity.
$$$ What I thought the Question Was $$$
A Class C of that age is likely to NOT have an Automatic Transfer Switch (ATS). ATS defaults your shore tie power cable to shore tie. If it senses the genny running, it switches to genny power. If you have an ATS it may be defective.
More likely, your Shore Tie cable is plugged into an outlet that's powered by your genny. YOU make the transfer to shore power when it's available by "unplugging from yourself" and plugging into the power pedestal.
It's a good idea to carry a 30A-to-20A (RV socket to ordinary household plug) and use it to check shore tie outlets before you plug in. You'll know if there's power and if it's wired correctly.
Sooo.... Do you know the shore tie you're plugging into is live?