Are you doing fusible applique or needle turn?
If you are doing needle turn you can modify it and take your applique pieces and place them right sides together with a white backing fabric such as lightweight bleached muslin or cheapy white solid broadcloth. Sew them together on your applique line but leave a bit unsewed to turn the piece right side out. Press the piece firmly and you have your applique ready to sew to your background piece and it is already turned under for you. The downside to this technique is the extra layer of fabric and the seam allowance from sewing the two pieces together may create a shadow, but I would think that would solve your shadowing problem of the background fabric and darker peices tucked under the snowman. Additionally a seam allowance shadow may be less obvious. Hope my explanation makes sense.