are you cross stitching on Aida cloth? or on regular cotton fabric???
if using regular cotton fabric you can add a stablizer (i usually just use a square of muslin the size of my block- put in hoop behind the main fabric- i stitch through both layers- keeps any strings from showing through and adds a bit of stability.
i would not use a fusable- it would make it more difficult to stitch through- they tend to add stiffness.
if you are using Aida cloth (cross-stitch fabric) you do not need to add anything behind it before stitching- but can add a (back-lining) after stitching. Aida cloth is quite (firm) and should hold it's shape without adding a layer.
there are some stablizers that are water soluble---wash away- but they are used for machine embroidery---working by hand eliminates some of the problems you face when working with the machine.