Altho not left handed, most people try placing their cutter down then cutting. My teacher taught me to place cutter, start cutting then cut into fabric. Come in at an angle. If you place too much pressure on the ruler, you will move it out of place. sut.
Of course you have to have your selvadges matched up and your fabric squared before you start cutting.
Another thought: have you ever tried to cut right handed? It may work.