my opinion is not that the basting is the problem, but with your sewing process. My first thought was to use a walking foot, but you already are. My other thoughts have already been mentioned as well. lessen your pressure of the foot on the fabric. make sure the quilt is supported on a table as you go. stitch slowly. reposition your hands and the quilt bulk frequently.
My other thought, if you are a new quilter, could this be an old machine? I had this problem in the beginning (though it didn't bother me like it is you!). When I bought a new machine, it was sooo much better. My first machine was given to me as a hand me down, and even though I had it serviced, it just didn't do well. I didn't know it wasn't good since I didn't know anything about sewing. My new machine was nothing fancy or expensive, but i found it so much smoother and cleaner to sew with.