If you need a fusible, my recommendation is MistyFuse. It is very soft. Does not come with a paper backing, but all you have to do is place a piece of parchment paper between it and your iron (or use a Teflon applique sheet when ironing).
With all fusibles, you will still want to machine stitch around the outside edges of the applique -- especially for a baby quilt, which will get frequent washing. However, you can do this with a straight stitch close to the edge.
Edit: MistyFuse is available on Amazon and eBay. Have not seen it in stores. Of the kind you can get in stores, Steam-a-Seam Light is the softest one I have tried. However, it is still much stiffer than MistyFuse.