Instead of pinning try binding clips - like snap hair clips, they are sold for quilting but I find them much easier than pins. Especially when sewing PUL, although not your issue here
Cut the minky and cotton at the same time, pref. with a rotary cutter on a cutting mat, and pin without moving it - that helps to not stretch the minky. Pin very close together.
You can get from Spotlight two kinds of spray - repositional applique and embroidery spray, and the slightly heavier basting spray. Both help hugely.
If you don't have a walking foot, try putting the stretchier fabric on the bottom so that the feed dogs help to move it through. Oops, just saw that that's opposite to what other people said - I guess try both and pick what works for you!
Hope everyone's tips help
Sewing minky (or even worse, coral fleece) makes me say bad words.