Are all your blocks the same size? That darn 1/4" seam can be a pain in the ???? (esp. in the beginner stage).
You say that the fabric are different, how 'stretchy' are some of them? The stretchy (aka polyester) one would be the one I worry about. In making your blocks, did you mix the cotton and polyester in the same block.
IMHO ~ if the blocks are the same size you 'might' not have to stabilize them. My reason that cotton around the 'stretch' fabric will help secure it.
Pictures are always helpful.