Your little monster is sure cute. I had to look twice though. He is pointing at something isn't he? heheheh!

I am making a quilt where all the blocks have very dense thick satin stitch applique plus machine embroidery stitches. Starch your fabric, use very,very firm wash away stabilizer or a strong paper behind your stitching. I find the stabilizer works a bit better while the paper is of course a lot cheaper. You are trying to get the fabric so strong that it won't be possible to pucker - about the strength of denim. Since you're using cotton, you have to get the strength from a combination of starch and backing. Using temporary spray adhesive to keep the paper on the back of your fabric is a trick that many use but it isn't truly frugal - works great though. Your block is fantastic!

I was told once there is a difference between stabilizer and pellon. Perhaps try pellon instead, I know one stretches more than the other. You could also be stitching too tight, too many stitches per inch, perhaps back that down. Perhaps also try a test run with a blanket stitch, I have always had good luck with it, and it lays nice and flat.

So cute! The one thing I learned was to use a light weight thread in the bobbin. The woman teaching the class said to use a thread called deco bob or lingere (spelling) bobbin thread. Also make sure you have the proper foot for satin stitching.

Try using an embroidery hoop. I have basted strips of fabric on the outside of my block if it is not big enough for the hoop. The hoop along with stabilizer should help.

I made the entire "Planetary Party" series (designed by Amy Bradley) and it was so much fun! I'm a "serious amateur" astronomer and I collected really funky fabrics for about 2 years. Because I divided the quilt in half for two wall hangings, rather than for a bed quilt, I used fusible fleece on each block's background before I appliqued. This would make it too stiff for a bed quilt that would be washed but it was great for the wall hanging. The "monsters" were fused onto that backing and I used a blanket stitch for the outlining. I must admit that I haven't mastered the satin stitch - yet.

I use stabilizer that I can leave in -- tearing it off seems to distort the fabric when I do it. Same with paper piecing -- I use the stuff that stays in.

Do you think that you could carefully cut the monster away from the backing (leaving the stitching) and then fuse the whole thing onto another piece of stabilized backing? You might need to use something like "fray check" on the satin stitching before you try it.

I haven't appliqued a block that large. Looking at the picture, it appears to me that the background fabric has stretched primarily when the satin stitch was on the bias of the background fabric and/or around tight curves.

I have not had this problem when I have done satin stitch applique; however, instead of using stabilizer I heavily starch the background fabric. My method is to mix a 1:1 solution of Sta-Flo liquid laundry starch and water, "paint" this solution on to the background fabric yardage until the fabric is saturated, toss in dryer, and iron with steam before I ever cut the background blocks (and it's always a good idea to cut the background blocks a little larger, then cut down to exact size after the applique is finished). Spray starch will not be heavy enough, even if you apply it in layers. The 1:1 solution I use is the heaviest starch application I have found to be usable.

You might want to do some test samples before committing to another large block. Some types of fusible may not want to stick to heavily starched fabric. What brand of fusible are you using? I find most fusibles leave the hand of the fabric too stiff for my liking. I am just in the process of testing Misty Fuse to see how it works with starched backgrounds, and also to see how it holds up with different types of applique stitching around the edge. No results yet.

Are you sure you want to use a satin stitch? The satin stitch alone creates kind of a stiff, bumpy edge. A slightly more relaxed zigzag might work better if you don't mind the stray raw edge threads that will probably develop from multiple washings. Actually, a raw edge type of applique might actually enhance the monsters -- making them look more "monstery".

Reduce the density of your satin stitching will help, they look very close together.

Have you tried blocking it and spraying with starch? It might be enough to straighten it out. Good luck.

Try using an additional layer of stabilizer. Wash away stabilizer would be great addition. It will give you the additional strength you need for the satin stitch.