Gosh, I never had this problem before except when I used a batting from JoAnns and HancockFabrics. Even the Warm&White from Hancocks seemed to have a little link. Therefore, I am sticking with my standby Warm&Natural (the darker one). Sure it is more expensive but if I spend all that time making a quilt, I do not want to spend time cleaning up and suffering from allergies caused from dust flying and lint clogging my needles, which has happened before. I have an 820Bernina and when I free motion, the lessor battings put out link like crazy. Maybe I need to slow down :)
As for the length of batting left before binding and during the machine quilting process of quilt, I generally clip down to 1/2 inch or so. I also long straight stitch at the extreme edge of my last border to keep it from stretching as I am quilting. Good luck with your cleanup and hopefully, you will post a photo of the quilt.