It took a few hours but I got the backlash fixed, or at least better. I was originally measuring about 10 thou on the x axis and about 15 on y. After a lot of measuring and fiddling I found it was due to a combination of preload on the lead screw bearings and backlash in the brass split nuts.
As shown in the picture below, the preload on the bearings comes from the #70 screw, there's no separate nut for it. After torquing the x and y handle screws to about 15 ft-lbs, using my calibrated arm, the backlash from the bearings came down to about 0.5 thou. I also noticed that these don't appear to be angular contact bearings, just regular radial bearings. After adding the preload the bearings feel a bit gritty and I don't think they will last all that long. If/when they do fail I'll try to replace them with some angular contact bearings.
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The other large source of backlash was the split nuts. They both needed to be tightened a pretty good amount. I didn't need to tighten them all the way but they're not that far from it. They're made from casted brass (which isn't a bad thing, these are cheaper to replace than lead screws) so not sure how long they will last before needing to be tightened again. These will probably eventually get replaced too.
After making all the adjustments I'm down to about 2-3 thou for both the x and y and the table feels a lot less loosy goosy, although a bit gritty because these radial bearings aren't meant for axial load. I'm going to cut some mild steel on it this weekend and we'll see how she handles climb milling now.