If you want a fusible that is not stiff, try Misty Fuse or Shade Fuse. These do not stiffen the fabric; however, you still need to sew over them to make them permanent for regular quilt use.
Stabilizer is different from fusible. Stabilizer is used underneath the background fabric when sewing an applique by machine -- especially when sewing the applique edges down with a satin stitch.