"Elaborate a bit, please. It feels like the bounce is coming from being on the balls of my feet a little too much for the Fox Trot. That, and some Fox Trot music simply makes me sway and bounce a bit more than strict ballroom fundamentals would call for or alllow."
The real problem is one of rushing through the action. Foxtrot type movement has a number of points in the action where it's comfortable to stay, and a number of points that feel insecure which are tempting to rush through. The problem is that if you rush through the insecure ones, then you hit the next part too fast, rise too high, etc and generally loose the character of the dance.
So you have to learn to move slowly and smoothly through those less secure parts of each step. Most of that will be developing strength in your feet, to allow you to move your weight further before you loose balance and have to go through to the next step. But also learning to take that between-steps energy and carry it through each step.
In the end your knee action will improve, but the knees are not really where the answer lies - the problem there is just a symptom.
Actually, the physically easiest way to do the dance properly is to make it fairly small, with very little knee usage at all and limited leg division, by doing it all with your feet. That has the additional benefit of developing foot strength. That's pretty much what you see even in competition videos of 20-30 years ago - proper and serene with outstanding control and mastery, but not overtly athletic in the way it looks today. The knees will only really start working when you scale up the distance traveled, and you don't want to do that until your feet can let you make those big movements very smooth and even instead of pulsing.