Breaststroke is very similar to the foundations of freestyle. When you are in the glide phase of the stroke you mentioned that there are points where you feel somewhat unstable and one side of your body is farther under water than the other. This is a result of being imbalanced and possibly have tension in your shoulders, neck, or torso.
The superman drill is great to focus on so that you can get a sense of being suspended by the water without any effort. The plus side of this drill is that its actually the gliding phase of the breaststroke which is pretty much the majority of your movement across the water. In the superman glide switch around with focal points. For example, first start off with focusing on a neutral head and see if keeping your head relaxed will help you feel that suspension and the feeling of your hips at the surface. Second, focus on relaxed extended arms. Make sure that extending your arms does not cause tension otherwise you will feel yourself wiggle to stay on that imaginary water balance beam. Its all about discovering what it feels like to be suspended effortlessly whether you need to relax more or reposition your arms or neck.