Welcome from Canada !! Just wanted to say that I too have had this problem & have just purchased something called a Slidelock (after being recommended here). It is NOT a ruler as it does not have measurements on it. It is a ruler "wannabe". It's a straight piece of acrylic with an attached handle with big grips on the bottom. With your ruler or the grid on your mat, you line up your fabric with those straight lines, put your Slidelock even with the lines, push down on it until it grips your fabric & cut away. Your fabric cannot move & apparently you can cut multiple folds of fabric with no movement & hence, straighter cuts. I have only just received it & have tried it only a couple of times but so far, so good. Fairly reasonably priced, so check it out
www.quiltersslidelock.com It may be your solution.