I too think that's very good for a first attempt.
What I do is heavily starch the background fabric before cutting the square (and I cut larger than necessary in case there is some distortion during sewing). Heavy starch provides enough stabilization so I do not have to add a stabilizer. My method for starching is to mix a 1:1 solution of Sta-Flo liquid laundry starch and water, "paint" it on with a large wall painting brush until the fabric is saturated, dry in dryer, then iron with steam.
I have not needed it so far, but if I felt the need for additional stabilizer I would simply add a layer of Stitch n Ditch (inexpensive very lightweight paper purchased from Amazon). This would be easy to remove afterwards and would not make the satin stitch any more stiff.
You might want to try out a couple of different stabilizers (including my starch method) on some practice blocks to find out which you like best.