I’m sorry about the trouble with your print results when running at higher speeds. Regarding your alignment, since your unit engraved first and then cut, the cut being misaligned is likely related to the shift in your engrave.
Could you check the tension of the belt below your laser arm for me?
When pressing lightly on one side of the belt, the other side should not move.
Like this:
(You should see an animated image above)
Let me know if the belt is loose and I will follow up with your next steps. Thank you!