The side to side movement of the crosshairs is parallex- no different than holding your finger in front of your view and moving your head side to side while looking at the find and the background behind it. This is not a good thing with scopes because you would need to have the exact same cheekweld every time if you want to shoot consistantly. I would call the manufacturer of the scope to find out how to correct that.
High-powered rifle scopes are set to be relatively parallax-free at 100 to 150 yards out to infinity. This means that objects viewed at ranges shorter than that for which it is set will show side-to-side and up-and-down movement such as you are experiencing as your eye is moved in relation to the ocular lense! The closer the object to your scope' s objective lense, the greater this movement seems to be.
I suspect there is nothing at all wrong with your scope, or the parallax setting it has! Unless your scope is equipped with an adjustable objective, (in which case, you need to set the objective for the minimum distance you will be shooting at), leave it alone. There' s nothing wrong, per your description of what you are seeing! Although it does not HURT to center your eye in the ocular lense window when shooting, it should not be necessary to do so, if the parallax adjustment is O.K. Shoot it on the range to verify this!
If you zero your rifle to be dead-on for windage at 100 yards, there is really no way to know whether or not it will be off windage-wise, or how much it might be off, at greater ranges without actually confirming this by shooting at the greater ranges! I have proven this to my satisfaction over the years! It' ll PROBABLY be off some, they usually are. It should not be off MUCH, but you SHOULD CONFIRM this with actual shooting!! Do this check on as calm a day as you can, to minimize the effects of wind. Or on a day when the wind is blowing straight up or down range.