I have this bug as well, Exceedingly annoying. I have found that the only solution to unstuck this glitch is to keep aiming and moving the left stick up and down/side to side, then the sights seem to unglitch and aim normally.
Can't believe that a bug like this can be allowed to exist for this long in a FPS game, the whole purpose is to be able to aim and shoot. Come on DICE.
