I started washing all my fabric before using it because I was getting a rash when I sewed. After investigating I found that not only does the new fabric have sizing, but it also has chemicals to prevent bugs from eating the fabric on the long trip from where it is made - and clothes made overseas do too. After I wash, I starch - heavily if I plan to use it in the accuquilt and lighter if I will be cutting with a rotary cutter.
In my clothing class I learned to pull a crossgrain thread so I knew where the grainline was so I could make sure things were straight on the grain. I still do this for scraps if it is important to have a straight grain cut.
Remember, for woven fabric the cross grain does not stretch but the lengthwise grain (the sides with the selvages) has some stretch. When fabric was wonky, we used to try to straighten the grain by having 2 folks hold opposite corners and pull, then move to the other corners (again opposite) and pull.