I used to cut fabric the way I think you are doing: you lay the fabric across the mat, with the cutting edge to your right and the bulk of the rest of the fabric folded on your left, slide the ruler on top of the fabric to your left until you have the right strip width and then cut your fabric on the right side of the ruler with the strip to the right of your ruler and the rest of the fabric under the ruler and to your left. You have to use the lines on the mat to do it this way, and you are simply using your ruler as a straight edge for the rotary cutter. Then one day I was watching a quilting video and the presenter cut a different way--using the ruler as the measure of the strip width. To do it this way, you straighten the edge of your fabric the way you usually do, but then FLIP the entire length of fabric over so that the new straight edge of the fabric is on the left side of the mat and the rest of the length of the fabric is on your right. Then place your ruler on top of the fabric, and then look through your clear ruler to line up the straight edge of the fabric under your ruler to the right strip width. If you are cutting a 2 1/2" strip, the right edge of your ruler should be 2 1/2" right of the end of the fabric--the strip you are cutting is UNDER the ruler. Remove that strip and keep going down the length of the fabric to your right. With this method, you are only using the ruler lines for vertical and horizontal measurements and don't look at the mat lines at all. I hope this makes sense. I am getting much more accurate cuts now with this method, but it took me a little while to adjust to it. Here is a photo of the placement of the fabric and ruler if I was cutting a 2 1/2" strip.
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