"My son needs to work on not moving or bouncing when he is hitting. "
Whoever told you this is repeating an old wives tale (not that there's anything wrong with old wives).
Your kid has the same flaws as most hitters.
His first move is to uncoil his torso, which opens him up too soon, and he starts his swing before toe touch (rear forearm is vertical), which leads to bat drag (hands behind the rear elbow).
1) As he moves forward (the stride) he must coil his hips and shoulders inward and hold this as long as possible before his front foot/toe lands. Practice this every day without a ball.
2) Pull the rear elbow back toward the catcher (like a bow) until he lands.
3) Never let his hands get behind his back elbow.
Do these three simple and free things and he will better than 99% of kids.