This is really super cute!!!
I haven't made a mug rug yet but I've made some coasters, and I gotta say, the most important bit for me to get the end shape looking right is making sure the batting is cut EXACTLY the shape I want the finished product, pinning the three layers together to help prevent shifting around as I sew (or if you're having big problems try a wash-off spray adhesive), and then sewing really slowly and carefully, using my perfectly cut batting to measure seam allowance. I watch the allowance from the batting rather than the top/bottom fabric because if you get wobbly around the batting, it gets bunchy and uneven. If you get wobbly around the fabric, you can just trim it