I never worry about straightening with the cross grain. I just straighten my quilting fabric by squaring it off on the end that I am cutting - in other words if the raw edges are not matching or off kiter, I trim it off so that both raw edges are the same. When quilting, I'm putting in enough seams and putting in enough actual quilting, that I've never had a problem with any twisting etc with my fabric.
I've only had that happen when making garments if I didn't straighten it with the cross grain and since I don't make garments any more, no worries.