I don't like tearing fabric because it damages the fibers along the tear -- about 2" in if examined under a microscope. Those microscopic tears weaken the fabric.
Being absolutely straight on grain is not necessary in quilting the way it is with dressmaking anyway.
I heavily starch the yardage before cutting background squares. Heavy starching stabilizes the fabric so it doesn't distort with handling. I cut the starched fabric squares larger than required and trim to size after the applique is finished.
Starching like this (including heavy starching of the background fabric) also helps prevent puckering when machine quilting.
In your case, I would recommend using spray basting as this keeps the 3 layers together with less slipping than safety pins. Also, I wonder how puffy your batting is. The higher the batting loft, the more likely you are to get the kind of distortion you describe after quilting.