Regarding stabilizers--if you are using the right one for your embroidery you should only need 1 layer. The more stitches require a need a medium to heavy cut away. If you don't want the backing to show, try a piece of organza. It is see through and will hold embroidery perfectly.
Also, the backing needs to be framed in the hoop with the fabric. It defeats the purpose the have it sitting underneath.
Hope this doesn't sound like a teacher but I use the excuse that I taught embroidery classes for some years and earned my living as a commercial embroiderer.
Wendy G.