You are correct that you have to re-square the cutting line on the ruler with most cutting methods. No matter how careful the fabric shifts/moves and goes off square.
Try this video.
http://quiltersplanner.com/power-cut...ay-cut-fabric/
This may be different from your method.
This is my favorite. (Found it again by luck for you.)
https://www.youtube.com/watch?v=NoBKOv3Azzw
Also try a larger square ruler instead of the strip ruler. It holds more of the fabric in place. Good luck!