I know it's a dumb question -- but I've made this mistake -- are your feed dogs down & is your stitch length set to 0?
I absolutely agree with Paper Princess about the pinning process being critical. Tape down your backing & pin from the center out every 4" or fist-width. The only other suggestion I have is to make sure you're not twisting or turning the quilt to create the circle. Your hands should form a triangle-like shape near the needle so you can have good control over moving the quilt. Just move a little bit, stop, re-position your hands & then start again. Don't pick your hands up while the machine is moving or that can cause puckers.
As far as it looking puffy like trapunto, I'm wondering what you have between your layers of fabric. You mentioned stabilizer. Depending on how thick that is & how thick your batting is, that could be the culprit. Circles do seem to puff up a bit more than other shapes even in most professional quilts I've seen. But if yours are puffing more maybe try adjusting the tension on your presser foot.