The "cheap" sights that came on Shadow's and 220's don't do that!
Anyway, I don't think the "side-to-side" is supposed to be controlled by the pin. The rear of the sight base (the part that screws to the gun) has a section that sticks up, and the upper part of the sight is supposed to straddle it snuggly. If there's too much clearance between this guide and the sides of the sight, it'll wiggle back and forth.
If yoy remove the elevation adjuster you can pivot the sight upward, and bend the side of the sight in a little bit so that it fits snuggly over the guide. The section you can bend is behind the "e" clip for the windage adjustment and ahead of the blade. If you try to bend the other side as wel, you're probably going to booger up the windage adjustment detent, so only do it on this side. The metal is quite soft, so don't go nuts.
And while the the sight is pivoted up, you can make sure that the screw that holds the whole assembly onto the rifle is tight.