Mitch,
It sounds like you are loosing all of the circuits that are inverter feed (inverter pass through circuits), and if so you have a faulty pass through relay in your inverter. To determine this, unplug from shore power, turn off the generator, and turn on the inverter. If the dead circuits now have power with the inverter working but no power on shore power and then you have checked the 30 amp inverter feed circuit to determine that there is 120 volt power to the inverter when on shore power, your pass through relay is bad. Another check is to check the 120 volt circuit at the inverter. if there is 120 volts at the shore power input on the inverter but no power at the inverter output when on shore power, your pass through relay is bad.
If you determine that you have a bad pass throigh relay, the correction will depend on what inverter you have.
Gerald