I press the fabric first, then fold it in half selvage-to-selvage. I keep the layers from touching at that point by keeping a finger between them (each hand). I shift the selvages until the fabric hangs straight and does not pull to either side. Then I allow the sides to touch, smooth it out and lay it on the cutting mat grid. Then I square off the edge.
Sometimes, I cut the single fold layer, sometimes I half it again along the fold and cut 4 layers. I have tendonitis, and cutting through multiple layers puts too much pressure on my system.
Often, I cut a large piece off that is a multiplier of the strip size. (Example: @2.5" strip, I cut 20" and cut that into 8 strips) This method seems to help keep things from shifting.