You definitely a larger needle to prevent the thread from shredding. About the needles breaking, you may be moving too fast or moving the needle too slow. If you move the fabric while the needle is still coming down, it will pull the needle and make it hit the plate. This may be caused by your speed, but it may also be caused by the sewing surface around the machine. If the room is humid the fabric can not slide smoothly over it and cause the needle to break. You may also want to check that the darning foot is not bent. Sometimes it may be a little bent and the needle hits is and that's why the problem doesn't happen when sewing a straight line. Trying to straighten it if it is obviously bent may break it. If that's the case you may need to remember to move the needle before sewing to avoid hitting it.
Pretend to sew with no thread or fabric and just watch the needle go up and down to see if it is hitting something. If is doesn't break try again with no thread but with a small practice sandwich. It may show where the problem is.
Hope this helps.