What most of the other posters said about stabilizer. I use a fairly heavy tear away stabilizer when I am doing satin stitch applique, and rarely if ever have puckering problems. Also, I experiment (most every time, because I forget from one experience to another) with needle placement and turning. For example, if I'm turning fabric to the right should my needle stop - before the turn - on the right or the left. One of these days I'll write it down after I again experiment....
and by the way, I think that your applique is lovely, and I am sure with some heavy duty pressing you can get those puckers under control.