Your chassis manual should tell you what the capacity of the cooling system is. You will not need to drain all the coolant but down below the thermostat housing. I would use an 8 gallon Rubbermaid tub. I would think drain about 4 or 5 gallons that would get you down below the housing.
The CAT C10 only has one thermostat but it very big and does not look anything like a car thermostat.
Another thing to consider is if there are booster pumps to move the coolant to the front of the coach. My coach has two booster pumps, one at the engine and one in the front. The pump help get the hot water to the front.
Also some HVAC systems have water cut off valves at the HVAC that can get stuck. Maybe they are stuck closed so no coolant is being allowed in the HVAC unit.
Do you have an accurate way to measure the coolant temperature, such as a ScanGuage or a SilverLeaf? You could also use an IR Temperature gun on the thermostat housing after you warn the engine up. If yours it is like mind you will have to go for a drive to get to operating temperature. Checking the temperature at the housing would give you a better idea of what is going on. You could also use the IR gun in the HVAC unit to determine coolant temperature there.
2000 Country Coach Magna
CAT 385HP Allison 6sp
1998 Saturn SW2