Many handicap golfers sway. I think you'll agree with that observation. In particular they sway the right hip to the right on the backswing. Sometimes this is because they have been told they should "shift the weight" to the right on the backswing and then to the left on the forward swing. Sometimes it may be just that they do it automatically as they try to swing the club back "on line."
Now I have an idea that may help eliminate the sway forever. I say "may," because we are all built differently. We can't tell whether another person feels the same way we do. Nor do we always appear to do what we feel we are doing anyway. But this idea has helped quite a few people I know:
The line of the backswing is to the right. But bodily it feels as though the force to the right meets a gearwheel so that it is immediately translated into a turn. It's like a rod pushing to the right against the tooth of a gear, so that instead of going to the right the force goes "round."
The initial force is backward to the right. But the result is a turn around the spine.
But as one swings down and through, the feel is that the force is exerted in a straight line to the left.
Some people translate this as "a turn around to the right on the backswing and a drive straight under on the downswing."
The hub however keeps still. It is more a universal joint. It sends the wheel round on the backswing, but down and then forward on the through swing.
See if this feeling helps you to conquer your sway.