I'm right handed and I used to have problems cutting with my rotary cutter. I decided that being nervous about cutting and wasting fabric was one reason. I would get shaky everytime I would make a cut. So I decided that I would slow down, breathe and forget about being afraid things wouldn't turn out perfectly. I have also discovered that not all new rotary blades are made with smooth edges. Some I have bought make jagged cuts almost chopping the fabric while other blades cut smoothly. I always buy Olfa and Fiskars.
I hope you figure out a way to cut your fabric. I recently purchased a June Tailor Shape Cutter on the advice of another quilter here and it has made a big improvement. Good luck. :)