The other thing that you should do when facing is to lock the carriage. Tighten the square head bolt on the right side of the carriage just to the left of the threading gauge. Also lock the compound slide by tightening the middle gib adjusting screw. On my 3996, I replaced one of the cross slide and one of the compound gib screws with a T-handle screw. The gib lock for the 10" and 12" milling attachment fits the cross slide. And the one from the 6" milling attachment fits the compound slide. Same locks will work on the 10" and all 12".
To get back to backlash in the cross feed, there are two sources for it. One is backlash or clearance between the cross feed screw and the cross feed nut. The other is end float of the cross feed screw. Before trying to determine or reduce the magnitude of the former, you should address the latter. To do this, first run the cross slide nut off of the end of the cross feed screw and move the slide another quarter inch or so to the rear. Then remove the two screw plugs in top of the carriage dovetail and squirt a few drops of oil into the holes while rotating the screw. Also apply a little oil to the rear of the dial where it runs near or against the front of the bearing. Replace the two screw plugs and run them down flush with the top of the dovetail. Loosen the dial set screw of thumb screw and confirm that the dial rotates freely.
Loosen the special nut retaining the crank. If you haven't done it recently, go ahead and remove all parts and inspect for wear. There are three original versions of how this all goes together, 10" and early 12", late 12" version 1, and late 12" version 2. In some cases it may be advisable to remove the bearing from the front edge of the carriage and face it off flat again. Version 1 and 2 of the late 12" have a flat washer (not in the same location) that may need to be either turned over or replaced. When all is OK, reassemble. By differential adjustments of the hex jam nut, 537-015 Nut or 049-089 Bushing, and the 10D-262 Special Nut, reduce the end float to as near to zero as possible without introducing appreciable drag when you rotate the screw with the crank. When you are satisfied with the adjustment, reaffirm that when loosened, the dial still rotates freely.
In several cases, this procedure has reduced the back lash when everything was reassembled by as much as one-half if you had never done it before.
Pull the cross slide back toward you while rotating the crank CCW. When the screw is fully through the nut, stop and determine the back lash using the installed dial. A new screw and nut would have on the order of 0.005". Assuming that end float is near zero, anything over that is wear in the nut. Crank the cross slide back to where it normally runs and re-check. Any increase over the reading at the end of the screw represents wear on the screw.
What you do from this point is up to you.