Firstly I agree with BrianB6, about the quality of the board. However, if it is what you have, then it is what you have.
I would check the outside edges of the sheet of ply to see if it is delaminating. If the edges are still sound, it should be OK. If the edges are splitting, then long term, any piece you cut may also split.
For cutting, I would try this. Tape both sides of cut location with tape, I prefer carpenters tape (this is the blue stuff that will not leave a residue). I would mark my line with pencil and then score the the cut line heavily with a sharp modelling knife or marking knife, before you saw. This will cut the fibers that are causing your surface splintering.
I hope this works for you.