Sharon, thanks for the tip. I'm having the same problem with my circles and now I know how to fix it!

I don't know how you managed those eyes, but they are way too cute! The owl is so whimsical that the eyes look perfect like they are! Love them!!

If you have a needle down function on your machine, it's a life saver on machine applique. It allows you to raise the presser foot and reposition the fabric as needed. Stop with the needle down on the outside side of the circle to reposition. Have you thought about doing a simple staight stitch about 1/8 in from the eyeball edge and allowing the edge to fray like in raw edge applique. It would be easier I think. If button eyes are put on make sure they are small to not pose a choking hazard for small children.They do look very cute!

Seeing what you are making, I would be tempted to straight stitch about an 1/8 in from the edge of the fabric circles and let it ravel out on both circles for the eyes and the same for the nose. Would make it look like lashes.
There are attachments that you can purchase for your machine to sew circles. I have also seen where someone used a thumbtack through a piece of tape, then taped to the bed of the machine so it is set up so the center of the circle is positioned on the pointed part of the tack and taped to the machines so the needle goes through the fabric an 1/8 in from the edge or what ever seam allowance you want. Hope this makes sense.
I must have been typing the same thing at the same time Tartan was.

CUTE! CUTE! CUTE! The imperfections add to the whimsy! I'm sorry for your frustration but I think the outcome is better than perfect would be. Just MHO.

How about treating the eyes and nose like raw edge applique? Think I'll try it and make a pin cushion out of it.

These are so cute. I'm going to try some. Thank you for the link.

Donna, this is what I do, draw your pattern on the fabric, do not cut it out,cut out all your appliques,place them on the pattern that you drew and glue them in place with glue stick, let dry then cut your pattern out, but not on the lines, in a large enough square that you can handle it as you zig-zag it on your machine. when all applique is done, I press the pattern piece, then cut it on the pattern lines and go from there.

I use flat metal washers, there are mylar products too that are flat circles, the hint is to pink your edge, this will reduce bulk which causes the crease.
Cut circle 1/4" larger than your washer, pink edge, spray starch, iron around washer, remove washer, repress.