Dimension accuracy loss under load

A good setup would be like this:
Stepper Motor
Good quality coupler
Either a commercial bearing with 2 opposed angular contact bearings or a double row angular contact bearing with a method to lock the bearing in the housing on the outer ring. Do not rely on the bearing just being a tight fit. The screw is machined in such a way that it’s locked axially at this end.
This is the fixed end.
The other end is best achieved with a commercial “floating” bearing block, this allows the other end to move ever so slightly axially whilst contain fixing it radially. The end is usually turned to a smaller diameter than the fixed end and a circlip holds the bearing on the screw.

A couple of notes.
The cheaper fk or bk blocks use simple roller bearings these will give trouble, the more expensive ones will have 2 angular contact bearings fitting in an opposed setup.
The couplers can have an affect, the spiral type are basically useless under load as the will “wind up” introducing backlash.
Do not use the stepper motor alone on the driven end, as per many 3D printers, nor expect then to contain the end of a screw via a solid coupler.
If you have a look at some ball screw catalogues they have some very good diagrams, even if not using ball screws the same applies.
Unless you have a specific reason to I would avoid having both ends of the screw fixed, temperature increases or decreases will cause the screw and the machine to expand and contract and you want one end of the screw to be able to move.
I’ve seen quite a few badly implemented designs that cause trouble for the owners.

Thanks for the information.
Picture I attached is the floating side. I can understand the need of bk and bf bearing for radial movement but my current issue is that the radial bearing housing is bigger than OD of bearing and under the load, the bearing wiggles within the housing. I checked both X and Y and they both have the same issue.

After some thought, I was able to remove nuts from the ball screw but to remove bearing from the housing may require whole disassembly of the machine.
For the issue that I can't cut precise round, so can't design the housing to fix the problem could be mitigated by using commercially available BF and BK blocks. Current machine design is that both X and Y plate has bearing housing hole cut out, so, I may have to redesign based on the bearing length.

BF & BK mounts mount on a vertical or horizontal plane, these are retangularish.
FF & FK require a round mounting hole and are located via 4 screws\bolts.

I wonder whether Loctite 660 type of radial gap filling as temporary solution is good idea. Once I fix this, I can cut out proper plates that can utilize FK/FF or BK/BF mounts.