Another point to check is the tailstock actually true to centerline of the lathe. So with the ram retracted line it up with the headstock now extend it fully and see if its still lined up, probably not. What's happening here is that if it's not lined up triue as you extend it, it will move out of alignment.
Another way to check it is with the tailstock locked to the bed, and ram fully extended and locked, set up a dial indicator on the carriage to track on the side of the ram, now wind the carriage the full length of the ram, the dial indicator will tell the sad story>
Another test while you're there loosen the ram lock and push/pull the ram side ways you'll see how much wear is in there. Then re-tighten the ram lock and slacken the tailstock/bed clamp, see how much movement is in there, be careful to not tilt the tailstock, just slide it sideways. and also twist it. This will show why you can not use the tailstock when setting up the headstock, it's just too unreliable, in fact it's just about the most unreliable part of alathe, you should always set it fresh each time you use it for anything important. If your tailstock is riding on a "V" bed then it should be pretty good in sideways movement, most of the wear will be confined to the ram. It's also pretty common for an older lathe for the tailstock to be below center, if it's over one thou, it can be shimmed.