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.
This is also happening to me on PC, happens very frequently. Makes playing very frustrating.
I don't know if there is deadzone sensivity on pc but setting it higher (13%) fixed my problem (it was 5% before).
