For a non-washable fabric, try steaming it and/or letting it sit in the sun for a while.
If you're using regular cotton fabrics, and other fabrics of similar weight, the stuffing makes a big difference, especially if you stitch through it a lot to condense it. If there isn't a lot of stitching, so the patch looks more like a pillow than a poptart, that stuffing won't help as much with structural integrity, but it still helps a little. No matter how many stitches are worked on fabric that lacks that quilting layer in the middle, the finished piece will likely still feel floppy and flimsy...Which is fine for many projects, but the stiffness is part of what makes these patches work as book covers and bags. Or, potentially, playsets
(scroll down on this link; such a great idea for a fabric dollhouse).
If you use thicker fabrics like fleece, fulled sweaters, or knit swatches, you'll be fine without stuffing.
My first several patches were stuffed with pieces of a fluffy bathrobe I'd cut apart for a project. Though that made the stuffing even, the fluff always pulled through with the stitches, and it looked bad.
For several years I've been trying to use up a bag of polyfil left over from a store-bought throw pillow that turned into a craft-fail
(I wanted some part of that pillow to go to good use!). It has not been ideal because it's very bunchy. I will try quaggy's tip to help with that. I compensate for the unevenness with tons of stitches.
For my last two patches, though, I inserted a scrap of soft, thick flannel between the two fabric pieces before sewing any of the sides. It's been so much easier to work with, and still sturdy. I don't know specifically what this fabric is called, but it is used for some baby blankets and other baby accessories; it isn't fleece (no fluff pulling up through the stitches), and it is very lightweight, but still thick. If I had quilt batting I'd definitely want to use that, but since I just work with what I have, these alternatives are fine.