I have also seen many turners leave the tailstock quill lock loose as they say it slows them down having to loosen and tighten each time you make an adjustment to the quill. Both of my lathes have slop in the quill and not locking it down could account for things not lining up properly and or moving as pressure is applied to the blank while turning.
My take on the quill lock is that it was determined decades ago that it was needed. If not, manufacturers would have done away with it as a cost cutting measure. It has a purpose so I use it.
Don't know if this your problem but just one more thing to check.
You could have several things combining to cause your issue. A little play between the bushing and the mandrel, bushing loose in the tube, bent mandrel, play between the mandrel and the mandrel saver, quill slop, head stock/tail stock alignment, tailstock to bed play, ends of blanks not being square, etc.....just a little off on several of these in conjunction can add up.
Needing to rotate the bushings confirms that something is not running true.
I only use between center bushings whenever I can. Even turning your blank close to the bushing and then going totally between centers for the last few thousandths, is another method to have concentric blanks. You just need a caliper to measure your blanks in comparison with the pen hardware. Remember to allow for sanding and also for finish.
Good luck!