"I know the knees and ankles play a very important role in this. How do we use them in creating a swing and sway
1) on a fwd 1/2 natural turn"
For the ankle, make sure you are starting to pick up your heel as your body passes over the foot on step one, which is to say before your free foot passes it. (This is a key element of the British standard tradition, which almost everyone else ignores.) In a waltz-type action, your knee will also soften forward as the body moves over the foot. Together these combine to send the center into the swing ahead of everything else - ahead of free leg, and ahead of shoulders. But of course the body must not be allowed to break at the waist - the resulting swing shape is very evenly distributed from head to toe. There is curve to the body in profile, but only a curvature to what is basically a straight line.
2) on a bck 1/2 natural turn
Toe in slightly while placing the first step, though keep the thighs closed. As your partner swings across the first step, your standing knee will veer inwards off the inside edge of your foot, such that you action towards step two is also almost a forward movement. Swing your center to match your partner's.