Ken, put a volt meter on the chassis battery and the coach batteries while at rest , should read about 12.6 volts.
Then plug in the MH to shore power and do the same voltage test , should then read about 13.4v or higher.
On some MH's, the MH converter will charge all the batteries, it will depend on how its wired. An isolator solenoid (which acts as a one-way electric gate) should protect your chassis battery from draining while boondocking etc. but that gate would open up to allow your converter to charge all the batteries.
If not, then you need to take measures to monitor the batteries not being charged.
Your 2011 Tioga should have a converter like a PD 4000 or PD 9000 series that has a charge wizard built into it. That CW monitors your batteries and kicks in the charge rate as needed while your MH is left plugged in to shore power for the winter.
Generator:
I would add some kind of fuel stabilizer to the MH fuel tank and start the generator once a week or every 2 weeks for 15 minutes or so. Doing that will allow the stabilized fuel to run through the carb and fuel line to prevent it from gumming up.
I find it best to drive the MH every other week and take it for a ride somewhere. It seems to keep the mice away , where they get into it to build a nest or eat the wires. If you don't use it, they tend to get a little too comfy unless you can keep them away somehow.
Use the proper amt of stabilizer of oz to gallons of fuel. A fuel stabilizer like StarTron works well to prevent ethanol gasoline from separating. We have also used SeaFoam in our boats and snowmobiles with equally good results for many years. We (at our marine dealership) tested about a dozen different brands of stabilizers and found only those two to NOT allow fuel separation in the test tubes we used for testing over a 6 month period.
If you can find non ethanol gasoline, that would be the best fuel to use with those stabilizers for long term storage but it's not always available. Call a boat dealer and ask about non ethanol gasoline dealers in your area.
