I used to appliqué my kids clothes in the manner when they were young, including a sweet little elephant once. I stabilized the appliqué piece with light, iron-on stabilizer before I cut it out. But no stabilizer on the background piece. Perhaps try a lighter thread so the ZZ stitch is not too dense. Or try lengthening your stitch a tad to open the ZZ just a little bit more. Maybe reducing the amount/density of the ZZ would smooth things out enough.