What kind of stitch are you using?
Mostly I do hand applique, but it sounds like you are doing it on a machine. With satin stitching, I just stitch over the beginning stitches with 3-4 overlapping stitches at the end when I've hit the start line again. With anything else, I pull my bobbin thread up to the top before I start stitching, knot off as soon as the threads start to get in my way & bury the knot in the back. For hand applique, I start with my knot at the back. For both, once I've finished, I pull my top thread to the back & knot the thread there.
I only use stabilizer when I am using a very dense stitch, like a satin stitch (or certain decorative stitches). Otherwise, it usually isn't necessary. The WonderUnder or wash-away fabric glue that I used to hold the applique in place is generally sufficient to keep things from distorting much with the applique. The only other thing I'd recommend is trimming your fabric a bit wider & longer for any blocks where you're going to be doing applique because it can shrink up the fabric a bit & there's nothing worse than having a heavily appliqued block that is .25" shorter than all the other blocks. I typically add .5", but I know some of the professional applique artists on YouTube say they add as much as 1-2" extra to length & width for any block that is at least 15" wide. Once the applique is completed, they square up their blocks to the originally intended size. I seem to recall Liuxin Newman ("Thimble Lady") does that on her applique work.