I was using a standard mandrel and went through the same thing. I presume I was warping the mandrel when tightening because it slowly got worse. At first I could feel it but not notice it in my pens. Eventually I could notice in my pens and replaced the mandrel system.
The problem went away. BUT, I bought the expensive live center precision mandrel system (not PSI) and two different sized shafts on the suggestion some bushing are loose, some are tight on a single shaft and that can cause similar issues. Lee's number 2 makes sense to be also because I was trying various squaring methods at the time and had a few flat but not square blanks.
In the end, I turn single barrels between centers. Still turn double barrels on the mandrel saver set up and the issue has went away. Oh, and keep sharpening/touching up your tools. Lessened my tendency to keep tightening the tailstock.
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