If this was just a utility quilt - one that's used for picnics or the kids drag around or we curl up with it on the couch but don't need anything fancy - I'd consider laying it on a new backing and tying it instead of quilting. Since you've already quilted it, you don't need to worry about how close together the ties are and could just do enough to keep the backing smooth. Then, I'd fold the backing to the front for the binding right over the top of the original binding. But if this is a quilt you want to preserve in it's original state, you'd be better off removing the quilting and binding and starting over again. Or, just remove the original binding, add a new back over the old, do minimal stitch in the ditch and add a new binding. Good luck.