To see if shorepower is providing current to the batteries, use your voltmeter (a required tool for all Rvers) at the house battery terminals while shorepower is connected and your house battery switch is on.
If you do not read 14.4 dcv then the Shore power, Power Cord, transfer switch, Inverter/Charger, Battery switch or big fuses between the Inverter and Battery is/are not working.
Start at the shore power source and measure output of each item listed above.
If you do read 14.4 volts at the battery terminal connectors, then the battery is likely toast.
You did not mention if running the Generator would charge the batteries. If it does, then the transfer switch may be frozen or you do not really have shore power entering the coach (check the source and the cord).
Do not assume anything when dealing with electricity flow.
1995 CC Magna #5280
C8.3L 300hp Cummins, 31,000lbs