some information on the character sway and movement on BF3 for you. Its most likely that your controller is the issue itself. It may just be getting old and be loose on the inside. Ive started to notice some random sway and movement, but i know its from my controller being worn out.
Fixing the sticks is no easy task. requires purchasing spare parts and soldering them... My advice is to buy a wired controller (if you would have enough length) as it also reduces the ping between controller and console.