The topic of sway has been a hot button topic for decades. I have observed an entire industry misunderstand Jimmy Ballard. He has been labeled as a sway teacher. The truth is that in any good golf swing there must be a Coil Behind the ball in the backswing. If you look at all the best ball strikers in the history of the game, they have all coiled behind the ball. Trying to keep your head steady or down will create a reverse pivot and not allow a person to coil. There is a way to sway, but it is drastically misunderstood.
To Coil properly on your backswing, make sure your weight stays on the inside of the trail foot going back. A great way to feel this is to turn your back foot inward towards the ball. This creates a tight coil and will not allow your weight to get outside of your trail foot. This is what a proper coil should feel like. Do this and you will be well on your way to making a great golf swing.