Clipping the seam allowance all around might help.
I think the easiest way to accomplish what you are trying to do would be to applique the circles onto the background fabric. Sewing circles is hard without a lot of pins or the right technique which only comes with practice. The quick way is to make a circle template, baste your fabric over the template with a gathering stitch, pull the gathers tight, use a bunch of starch to hold it, remove the template, applique to background fabric. You could use freezer paper for the template and sew the circle down then carefully snip open the back to remove the template. Freezer paper will help to stablize it while you stitch it down.
I've also seen a funky inside out method of making a circle on a square that is pieced, not appliqued. Simply Quilts, maybe?