While I have no specific knowledge of the specifics of your unit. (Actually I donít see a reference to how old a unit you have). But I would check the other components in the system besides the batteries:
1/Output from converter measured at converter (assuming converter based system rather than inverter-based)
2/Same input at battery end to confirm good cable (ground and line)
3/Battery isolator. (Why is chassis battery charging and house battery not)
4/ without shore power and with engine running, measure #2 above again to see if house battery is getting a charge while driving.
It seems strange to me that your chassis battery is getting a charge while on shore power, unless you have a modern electronic version of the isolator. Most commonly on older systems the chassis battery does not charge from the converter so many of us run a separate battery tender during Rv storage to keep the chassis battery topped up. If you have older style (tin can) isolator, maybe it is connected incorrectly?