The batteries should last forever when you're plugged in, because the converter is supposed to provide the power to charge them and run all of the 12 volt stuff when you're plugged into power or running the generator.
It sounds like your converter isn't working, so whether or not you're plugged into shore power the batteries are running your 12 volt stuff, until the batteries run out.
If the A/C has a wall thermostat, it needs 12 volts to run the thermostat and the A/C's control circuits. No 12 volts - the A/C stays off.
Your refrigerator also needs 12 volts to run it's control board. No 12 volts - the refrigerator stays off (even in gas mode).
Turning on the ignition connects the house and starting batteries together, so the house 12 volt stuff will work ... until you drain the starting battery.
You really need a multimeter to see if your converter is working and charging the batteries. We can talk you through using it. If there's a Wal-Mart nearby, they have one in their tool section for about $25. Or go to Harbor Freight, they have several.
One thing you can try first ... the two 40 Amp fuses on the left side of the 12 volt fusepanel as shown in your picture are Reverse Polarity Fuses. They will blow to protect the converter if the battery is accidentally connected backwards even for an instant. They don't have the Blown Fuse lights like the other fuses so you'll have to pull them to see if they're blown. You should see the conductor going from one pin to the other when you look through them from the side - if it's missing a piece or not there it's blown.
And here's a link to your converter's Service Manual if you or someone else starts troubleshooting it:
https://wfcoelectronics.com/wp-content/uploads/2019/06/8900-Series-Manual-web.pdf