G4L,
No way, those animals aren't standing still for you? They're probably just trying to get out of the wind!
Though I practice much more wingshooting than I do with a rifle, it's only been the last few years and I still swing a shotgun the same way I swing a rifle (often resulting in a miss).
The last two deer I shot as well as the last elk were running, though the range was under 100 yards on all three. I can't scientifically say what the secret was (all three were dead on their feet), other than 1) confidence in my shooting skills, and 2) steady swing from behind with let-off (that's the catch) happening right about the time the crosshair passed ahead of the shoulder.
Good luck.