The alternator should charge the batteries-- easy test with a digital voltmeter:
Check voltage at the battery (chassis and then house)with engine off, shore power and generator off. 12.7 or more is a fully charged battery. 12.2 is 50% discharged.
Start engine and run at high idle (like 1200 RPM). Recheck voltages at battery. Assuming batteries not deeply discharged, voltage should be 13.7-14.2
Turn off engine and turn on some loads to dissipate surface charge. Plug in to shore power and check voltage at the house battery. If smart charger, should start around 14, if not smart one, about 13.5.
I do not know if the chassis battery charges when on shore power on your coach-- Winnebago can tell you if the manual does not.