"Bouncy" action of the body is often seen in newer students of American style Rhythm, who place too much emphasis on the bent leg action.
When you commit all of your weight to a bent leg, particularly with a flat foot, you will tend to drop your center. This is a very common mistake in beginners. When the leg straightens, your center will pick up again, and you'll get the bouncy action.
So the first thing to realize is that the level of your center is basically in a normal or "neutral" position when you are standing upright on a straight leg. In order to avoid bouncing, you must maintain your body at this level as you bend and straighten.
Try this: Stand in first position with your feet flat and your body weight over your right foot. Make sure that you are standing tall and lifted, hips "neutral" (not settled to left or right). Now go ahead and bend your left knee, making sure that it does not have any body weight, and allowing the hips to settle rightward.
Here's the important part: As the hip is settling, pay very close attention to your posture and particularly the height of your sternum (place a finger on the sternum to really draw your attention to it). You should not feel yourself pitch forward or downward. The sternum should be able to maintain a picked up position, and the position of your upper body (specifically the chest and shoulders) should not be afftected by the setting action.
By focusing on the picking up of your center, sternum, and upper body, especially during the settling action (where it's most likely to get lost), you can teach yourself how to maintain a nice, level position during hip action.
As an exercise, try standing in first position and settling the hips from one side to the other. For now, always shift the weight to the straight leg. As noted above, always pay close attention to the level and position of the upper body.
This exercise can (and probably should) also be done in 2nd position, 4th right foot forward, and 4th left foot forward. Once you get the idea, you can start to focus on applying it to actual movements and dance steps, where there are further issues to tackle. But for now start with this, and see how it goes for you.
In the future, we can discuss the more complex issues such as the timing of the bending/straightening action with respect to the steps, as well as the timing of the lowering of the heels.
I hope this helps.
Sincerely,
Jonathan Atkinson