I think its normal that it "shakes" when you move it fast arround. It is just realistic. If you move it slower it does not "shake"
Did you watch the video though? The gun is literally shaking when I look around, not just swaying with the movement. I know this isn't intended because it only happens in the campaign of this game and not in any other call of duty game, whether it be singleplayer or multiplayer. Also, it shakes regardless of speed.