There are several keys to NOT SWAYING!
First pre-swing–be sure to not get too wide of a stance. We all have heard shoulder width–but your feet are not attached to your shoulders–they are attached to your hips. Your stance should never get much wider than hip width with a driver and less with the irons. A too wide stance makes it hard to shift the weight without swaying. Second be sure to have a slight tilt in the spine toward your trail side. Picture included–I call this hip bump/spine tilt, slightly bump the hips toward the target and let the spine tilt away from the target, your lead side hip and shoulder should be higher than the trail side joints.
In swing–this is KEY! Turn the trail side shoulder away from the ball and toward your spine and behind your head and at the same time feel like you are turning the trail side hip and butt cheek toward the target. This will keep you centered over the ball and allow you to maintain the spine tilt you created at address.
Here are some good check points. 1. Your trail side foot should stay flat on the ground. 2. Your trail side knee should stay inside your trail side foot. 3. Your head may move “off” the ball some but not a lot. 4. Your weight should be toward the trail side heel.
All good players LOAD into their trail side and do so without swaying–if you sway you are not getting loaded correctly and will make it very tough to get back into a proper impact position.