Omnigrid puts out a clear plastic product called InvisiGrip that I have on all of my rulers. It's clear so it doesn't interfere with seeing the ruler markings, and it just clings to the rulers so no sticky stuff to worry about. It really does keep the rulers from slipping. A 1 yd roll did a dozen of my rulers in various sizes and shapes. I use a couple of the Olipfa rulers too, and the lip does help.
In my travels, I came across one tip that I can't remember where I heard it from but it has helped a lot - choose your cutting mat lines, or the ruler lines but never mix the two. If they vary, the cuts will be off. I keep the piece I'm cutting under the ruler, and to keep myself from getting confused by the lines on the mat I just turned it over.
I had problems getting straight cuts, and then I watched someone straighten the fabric by matching the selvages and letting the fabric hang in the air to see if it was straight along the fold (and then sliding the selvages along one another until it was straight.) It was new to me! I started doing that, and discovered that all fabric is not created equal - some of my fabric was shockingly distorted when I tried to straighten it this way. I can't imagine getting a straight cut out of wonky fabric.