For what it is worth, I am not getting notified when someone responds on this Forum either. I had the exact same problem with my X and Y axis. Dave Clements from Arizona CNC fixed both of them for small $$. You may want to reach out to him to fix yours. Both my X and Y axis have less than .001 backlash. I did not have Dave re-work my Z axis and it has .004 backlash. Unfortunately, it would take lots of effort to remove it from the mill at this time.
I have been following an online CAM course (Titans of CNC Building Blocks) and have machined 8 of the10 parts so far with really great results. I am about to run a part that will certainly push the limits of the Z axis. If the .004 backlash cannot be compensated with the backlash compensation, it will be very visible on this part. Here is the part I am about to run:
I will keep you posted as to how it works out.
Best… Richard