You need to understand how your rig is built. The vast majority of the weight sits on the axles and tires. They give / allow movement by design.
Unless you want to out blocks under the frame and remove the wheels the rig is going to have some movement when weight shifts around.
All the suggestions above will help minimize that movement. But the key is to get weight off the suspension. Everyone has a different level of tolerance for movement.
Try one thing first, but do not expect it to be a cure all. You will need a couple dozen nights in the rig to determine what works for you.
Time taken to get the rig setup and properly level helps.