It'll help to practice throwing your punches while circling mindful of the timing between your punches and footwork. For example let's say you're circling right, as you move your left foot jab at the same time. Now after you pivot (or when you're facing your opponent) throw a straight right. You can repeat it (jab, right, jab, right...) so that it becomes natural without needing to hit the breaks to punch. Also practice circling both ways as that's what you're going to have to do when you get in the ring.