When doing needle turn appliqué, I tend to scrunch/stretch my background fabric resulting in a wrinkly background. I have found putting my background in a non-slip embroidery hoop helps me. I then pin my shape in place and no matter how I sew the shape in place, the background doesn't pucker.
With doing such a tight spiral, you are going to have to clip the edges of the seam allowance so that the seam allowance can open a bit in the back. I leave about 2 threads from the edges when clipping.