I was taught to not only iron but use spray starch on fabric before cutting.
It is more work but my fabric is stiffer therefore feels like it "moves" less when folding and cutting.
I have never ripped material always have cut. Was reading in directions for a log cabin quilt to line up material selvage to selvage and re line up every 3 or 4 cuts.
Also how many layers are you cutting at once? Usually 3-4 is best so they don't slip around. one seems too thin to stay in place. more than that requires too much pressure to cut
hope it helps :-) Marybeth