Instead of sandwiching the sleeve between the cuff layers, try this instead:  Make a contrast piping and sew between the cuff layers to finish the end of the cuff.  Then sew the sleeve fabric layer to the end of the sleeve, keeping the cuff layer free.  Turn under the seam allowance on the raw edge of the cuff and hand-sew over the seam on the lining side.  The piping makes a more elegant finish than binding.