Have you considered cutting your fabric in lengthwise strips (parallel to the selvage) instead of crosswise?
When I cut my strips lengthwise, I at least will have two edges on grain.
I also do better with only a couple of layers at a time. Might take me a bit longer to cut stuff, but if I do goof up, there is less fabric wasted.
The other suggestions to make your ruler more slip resistant are definitely worth following.