I think you are confusing fusible for stabilizer. Fusible is the stuff that sticks the applique to the background fabric. Stabilizer is what is needed underneath the background fabric to prevent puckering while you machine stitch the applique.

Misty Fuse and Shade Fuse are fusibles that do not stiffen the fabric. I have been waiting for some time to do some Misty Fuse samples because I want to see how it holds up in the wash with different stitchings around the applique.

The background fabric needs to be stabilized so it doesn't distort while you are sewing around applique shapes. What I do is heavily starch the background fabric with a 1:1 solution of Sta-Flo liquid laundry starch and water before ever cutting the background pieces. I "paint" this solution onto the yardage using a large wall painting brush until the fabric is saturated, toss in the dryer, then iron with steam. The fabric comes out fairly stiff with this method. In any case, typically it provides enough stability so the background fabric does not distort while you are sewing. If you do not use this heavy starching method, or if starch is not enough, then you need to place something underneath the background fabric before sewing the applique. This can be a tear-away stabilizer or a water-soluble stabilizer. Sometimes the stabilizer needs to be doubled, and I think it is more effective if you can spray baste it to the background fabric so the fabric is unable to slip away from the stabilizer as you sew (although most people do not do this).

Do you satin stitch or button hole stitch your appliques in place. I am a bit concerned about fraying etc without the fusible to hold it together.

I like Misty fuse. its very light weight and soft.