I like the theory behind mandrel savers, and have tried two different ones. One from PSI, and a higher quality one from CSUSA. Neither one worked for me. I couldn't get the blank tight enough, it would always spin during turning. Now granted, I'm often too aggressive and impatient, and try to take too much material off during each pass. Regardless, mandrel savers just don't work for me. I have tried them both multiple times over the years, always with the same result.
Having said that, I know they do work well for some turners. As to your problem with the mandrel wobbling, I would say either the mandrel is too thin or the bore in your mandrel saver is too large, or there's an issue with either or both Morse tapers on your lathe. I have had issues in the past with the Morse taper bore getting dirty in the headstock, causing the mandrel to wobble. I cleaned it with a twisted rag and solvent.
And I don't see how tightening your tailstock too much could bend a mandrel with a saver. It doesn't put any pressure on the mandrel itself, it puts the pressure on the blank(s).