I am a relative inexperienced appliquer and feel the stabilizer makes my stitches look a little smoother.

I just use copy paper as stabilizer. Much can be torn or cut away, and I soak the embroideries in hotter soapy water for several hours, even overnight. Then use a smaller scrubbing brush to scrub away the tiny bits.

all types of stabilizers can be bought from Jenny Haskins or I think RNK are the US distributers.

I sometimes do huge (bed size) applique) with a 30 wt. Mettler thread and very tight satin stitch. I use wall paper liner for my stabilizer otherwise the stitches can pull too tight and cause tunneling or not lie smooth enough and cause problems. Usually the stitches cause the liner to fall off as I am working leaving the piece under the stitches which I leave in with no problems. Here is the backside of a 48 inch rooster in progress.

It can make them look nice, I agree. You can find wash away at Joanns. They have a few different kinds, for light or heavier designs. I buy it on the bolt and it doesn't cost too much. this is the way to go if your zigzag stitches aren't close enough together to cover the whole thing.