Wadding Pindots!

Hi, I have a problem with the backing on a quilt. A lady at a local sewing shop recommended the size 11 needle for my stippling, so I gave it a try. Half way through the quilting I noticed the tiny white pindots on the back. My cotton, [top and bottom] is black so I was a bit stumped when I saw the white! It seems the needle was too fine, and it pierced the fabrics and pulled the wadding right through to the back! I am trying to finish the binding so that I can give it a wash, hoping that might fix the problem. If anyone has had this happen to them, can you let me know if it is fixable? It's a 50th birthday present and I do want it to look good. My last resort will be to take binding off and turn it into a quillo. thanks, Gail