I machine appliqué a lot. I also use invisible thread a lot. I use regular thread in my bobbin in a neutral color. I use Heat n Bond lite as my fusible, haven’t had a lot of luck with steam a seam.
sometimes using a lightweight stabilizer on the back helps - especially if the background fabric is very lightweight. I Always use a blanket stitch to stitch appliqués. The straight horizontal stitch laying just outside along the edges of the appliqué, the ( bites going in about an 1/8”. Zigzags always cause fraying and I am not a fan of satin stitch at all. A small blanket stitch is pretty much invisible on the finished pieces.

I usually use the blanket stitch on my machine applique and it seems to hold down better without little threads peeking out. You may also consider making you zig zag stitch closer together.