Before re-quilting on your quilt I would make up a test quilt sandwich and sew a few lines and see what happens. If it is the thread, get rid of it - or save it for basting only. I can't image it being the tension without seeing it in the quilting with lots of puckering. You can try different thread and see if it keeps happening.
Are you using the same thread in the bobbin as you use on the top? Some people say that polyester thread will cut cotton thread. I have not had this problem. I did have some very old thread do it - but it was over 70 years old and that doesn't seem to be the situation here.