When you turn between centers make sure you are using a soft center that you turned on your lathe , do not remove it from chuck, or you will have to retrue it, not a hard job though. If after doing all you can to adjust carriage and dialing in tailstock you should be able to hold .001 even on a very worn machine. I believe there are posts at PM describing how to check for bed wear. When the tailstocks are new and scraped in for that particular machine, they are a tad high, a few thou and point down slightly, this gives you a margin for wear, as time passes most folks end up shimming between base of tailstock and top piece. On a 40 year old lathe I suspect your tailstock is going to need some TLC unless this machine was a one owner run by the person who owned it. I urge you to spend a lot of time at PM monarch board reading. Lots of folks have been down this road before you, only a very few are on HM, there are guys way smarter and more experienced than I at PM, Donie, rkepler, J Tiers, and many others.
michael