After sending the patches through the washer and dryer a few times (because if they’re going to shrink, I want them to shrink *before* I sew them on my sweater thank you very much), I pinned them to the sweater. Then I held the fabric stiff and still with an embroidery hoop and started sewing. I picked a column of stitches on the edge of the patch and duplicate stitched over every other stitch in the column, then over a random scattering of stitches in the middle, just so the patch and the underlying fabric would be held together.
Once the first one was done I did it again, making at least a token effort to line the two patches up so they happened at the same place.
More pinning, more stitching, and in no time they were both on there. And now I have a sweater that should happily last another year or two at least.
This is the bit where I tell you, fairly sternly, that you can do this too if you want to. It’s not hard (make new fabric where the fabric is missing, reinforce spots where the fabric is getting thin or frayed). You can google ‘how to mend’ if you’re feeling really nervous, but really I think it’s probably safe to just jump in and experiment. Almost everything you’d do as part of mending is reversible (you can pick it all right back out if you don’t like how it’s coming out), and if you were going to throw the piece away anyways, you’ve got very little to lose. Plus when it works out you get to feel absurdly smug.