I am using a moistened toothpick to push under the fabric as I'm working through the needleturn appliqué but just remembered something my mom had shown me last year when she was giving me the hands-on "basics" lesson and it really works well for me so I thought I'd share it: with the toothpick at the edge of the appliqué design line, I lightly twirl the moistened toothpick as I'm pushing the fabric down. The twirl motion pushes the fabric under smoothly and there is less distortion of the fabric and fewer puckers.
Also, I think Sherry alluded to wanting to look at how Liuxin Newman holds the needle after I mentioned that is what I was doing. I simply hold the needle as most of us do, between the index finger and thumb with the end of the needle gently resting against the middle finger. The important thing here for me, is to ensure I put the needle perpendicular to the fabric as it goes down into the background, then I bring it up on a slant like everyone else does.
Finally, my mom taught me to bring the needle up and pull the thread away from me, not towards me when the stitch is completed: this will make it less likely that your thread will twist.
Hope this is helpful to someone. They are small details but I'm finding that in needleturn appliqué, especially since I'm doing the half-size quilt, details can matter. Back to my needleturn!