My favorite method is to starch the background fabric heavily before cutting the squares. I use a 1:1 solution of Sta-Flo liquid laundry starch and water, saturate the fabric, toss in dryer, then iron with steam. This makes a very heavy starch application. I also cut my background fabrics a little larger, just in case the sewing takes up some fabric, then cut to exact size after sewing.
Since you are already sewing, the starch method won't work. You will need to place a stabilizer underneath the background square. A plain sheet of typing paper would probably work, or Stitch and Ditch (inexpensive on Amazon and tears away easily), or a tear-away stabilizer. MistyFuse underneath would probably work too, but I haven't tried that.
Basically you need to stabilize the background fabric so it doesn't stretch and/or tunnel when you satin stitch. Applique requires you to sew all around a design, so you are inevitably going to be sewing on the bias of the background at certain points. That is where the background fabric is most likely to stretch and distort. Stabilizing the background fabric prevents that.