If you have that much end shake, Your going to need several shim. Cylinder shouldn't move forward and back over .002. .003 at most.
This will leave a big Cylinder to barrel gap. Which needs to be fixed by removing barrel, trimming shoulder enough that it turns in another another complete turn. This will move the barrel in .028. Then after reinstalling the barrel shank will need trimmed to adjust the cylinder for appox .006 gap.
But, before you sag your head. Check that the yoke isn't moving in and out. A loose yoke will cause cylinder movement also. If the yoke moves, remove the screw and look at the groove and "button" on the end of piece that slides in frame. The end of the screw may be worn and maybe the button side of the groove. A new yoke screw will help and then using a small smooth hammer tap the area of the button where it rides on screw will move metal over to tighten up the groove. Then, the new screw and the adjusted button will hold yoke all the way back tight in the frame. Don't wack it just kind of massage it with the hammer.
If your yoke has movement adjust that out before adding shims to yoke tube/cylinder for end shake.
I almost bet is a combination of yoke movement and end shake thats giving you thatt much movement and the big gap. Hard to believe a 32 S&W long would shoot itself loose. Every time the cylinder rotates there is a tiny tiny bit of wear, every time the cylinder opens and closes there is wear on the screw and the button and while joggling around and loading rounds the cylinder is pressing yoke button on the screw tip
