Sounds like your house batteries need replacing. Easy enough to test...get a volt meter and check the voltage without being connected to shore power and with the engine not running. You should optimally be getting somewhere around 12.5V. If you don't have that, then likely your battery is dead or dying. If you do have that, then try to start the genset while checking the voltage. It shouldn't drop to less than maybe 9V while the genset is turning over and then recover when you let go of the starter. If it goes under 6V and doesn't recover, your house batteries are likely dead or dying. Plug in to shore power and then check again....you should be somewhere around 13.5-14.5 if charging.
Again, my bet would be the batteries need replacing.