I neglected to mention some things that I DID do regarding a lead screw/split nut alignment issue I had. I did the full alignment procedure that's been mentioned above and still had a pretty significant misalignment problem. One issue was that the alignment always shifted when I tightened down the screws that attach the apron to the saddle. I found that the bottom faces of the original screw heads were rounded so the screws always moved the apron around -- so each screw was centered in the saddle's slotted holes. Buying better screws fixed that problem, but there still was some residual misalignment.
I don't have any photos of what I did next, sorry about that. If there's interest it would be fairly easy to at least re-create the setups I used.
The next thing I did was to refine the right-hand pillow block (my observations showed it was the main culprit). I made sure the surfaces underneath the mounting screws were flat (using my mill), and I also milled the bottom of the pillow block to make sure it was flat. To ensure the bottom plane of the block was exactly parallel to the borehole (where the leadscrew goes), I made a mandrel to hold the pillow block and used my lathe to face both ends of the block. This gave me two sides that were parallel to each other and exactly perpendicular to the borehole. I used those reference surfaces to set up the milling job for the bottom of the pillow block. I had thought I would need to add some shims to get the correct alignment but lady luck was in my favor that day -- I removed just the right amount to get the alignment right! That was a first, for sure.