Well in MM mode the Avid has 48db of gain, which is very high for MM, which is typically around 40db, and the Ortofons tend to measure high output, so what you are hearing could be a mismatch. In which case the only options are to live with it, change your phono pre, or cartridge.
But let's eliminate another possibility before you head down that path. Is it possible that the Avid is just the clearer and allowing you to hear distortion in your setup that the Pro-ject does not allow you to hear? First thing that I would do is double check my alignment, radical stylus cuts like the Shibata and fine line, require exact alignment. And this includes VTA, so I would ask how you are adjusting VTA on your turntables?
A little experiment, and it is a temporary one, is to increase tracking force by .2 to .5 grams (2 to 5 tenths). If the distortion changes in character it is probably your set-up and alignment as the output of the cartridge is still the same. If the distortion does not change, you may have a mismatch. Good luck and get back with your results.