mushy ride

Towing a car behind your motor home adds no weight to the motor homes chassis. What you are feeling is the car pushing and pulling you around. Do you have brakes on the car or dolly that work with the coaches brakes?? If not, you might consider having them hooked up, it will help with slowing down and stopping, it won't push you around.

It almost sounds like you might be towing too much car for your motorhome. We cannot tell ours is back there but it is only 2800 pounds so does not push us around. As a rule of thumb a gas MH can tow up to 5K pounds while a Diesel is 10K.

One thing to note, where the tow bar connects to the car it should be level to no more than six inches below the coach connection. If it is too high it may affect the way the coach feels on bumps and dips.

The six inch number is from the Blue Ox manual.

The car only weighs about 3100 lbs.
The car connects about 1 inch below.
I stopped at a shop and they recommended checking the airbag stablizers.