It's very easy to make clean cuts on prints adhesive mounted to Gator, as opposed to canvas on Gator which is much harder.
My best recommendation is to sacrifice a mounted scrap print and board to the cause of learning the trade. Just keep slicing. Education is always worth the price.
For medium to heavy weight papers, what works best for me is to rotate the knife very slightly in towards the print, make perhaps two gentle cuts to get through the paper, and then aggressively cut through the rest of the board. That should yield a smooth edge on both the foam and the back laminate. If you see fractures on the back laminate, take a few medium cuts instead of a single aggressive one.
The required skill is keeping the knife at the same angle for both gentle and aggressive cuts. It's all about knife control, and that's why one should practice. Virtuoso Gator cutters are known to practice several hours a day.
The purpose of the angle and the initial gentle cuts is to cut through the paper while generating the smallest possible burrs. The proper angle will place a large burr on the waste piece, and a relatively smaller burr on the print side. You can never quite lose a burr on the print side, but it can be gently burnished down by the smooth chrome rod of a screwdriver, smaller burrs are easier.
If thin, long shards of paper are generated from the cuts, it's because the knife position was changed for sequential cuts. That's the argument for a single aggressive cut, which will unfortunately generates a large burr on both sides of the cut and sometimes fractures the back edges of the Gator.
A cutting straight edge like
this is a must-have, IMHO. Those have wide, foam covered bases that can be safely placed over the most delicate prints. I've got two of those, and a table surface completely covered with an always present cutting matte that is a suitable surface for every type of the print and frame handling, and I laugh at rotary cutters.