Well you should be able to move it with a good hard push. If it is that tight you will get stick slip and that is never good.

Clean and adjust the ways. I can't see them but there must be a way to do it.

Pull the Y motor and turn the screw by hand. If you have no resistance when you turn the screw for almost any rotation then that is the screw nut or end bearing slop or ball mount.

You can pull the ball screw and re pack the balls for less than $50 but you need to know the exact size of the current balls. It is fairly easy to do the packing part but you really want to know that it is the nut first.

Bal-tecâ„¢, Manufacturer of Precision Balls, Ball Bars, and Kinematic Components has all the balls and can measure what you have. You will only need to go .0003" or so larger unless it is really loose which means it might be trashed.