My guess is that if kicking the right knee in fixes it, it's not a sway that's causing it but an insufficient hip rotation. Kicking the right knee in gives the hip a tiny headstart and that's why it works. It's probably as good a fix as any, and I've seen "swing tips" that recommend doing just what you're doing, "cocking" the right knee toward the target a bit. I find that angling the right foot in a bit or just holding it dead square and not letting it flare out at all does the same tng, but that may be because I'm duckfooted to begin with.

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Todd
Philadelphia, PA
USA

The reason the pro tells you to keep your head down is so you can't see him laughing. ~Phyllis Diller