The stuff you really need is called interfacing. Stabilizer mainly is used in embroidery machines. Interfacing like stabilizer comes in various weights. You should look for a "mid to light weight" interfacing. Not the kind for cuffs-collars - it will be much to stiff. You can sort of tell if it's the right stuff because it will glide over your hands and doesn't have much body to it. More of a drape.
Now... If you want to "deal" and get through the stuff you bought you can go ahead and use it, sew it in like you were planning to and then once the quilt top is done you can "tear-away" the stabilizer that way the fabrics in your quilt wont be "stiff" forever. I made the exact mistake you're talking about in the baby clothes quilt in my avatar, but I'd already spent so much money in stabilizers and interfacings that I just dealt with the blocks that made with the wrong stuff.