If your quilt sandwich still has one movement, stick with the vertical lines and just add more of them. If you add horizontal lines you will sew in puckers in the back where the lines cross. As for keeping your lines straight, painter's of masking tape can be stuck on the top and you can sew along the edge of it. I use it more then once and just add a few straight pins to it when it stops sticking.
P.S. Welcome from Ontario, Canada and cute quilt!