What if you were to continue to breath on your "good" side, just to get the oxygen, but turn your head as if you were planning to breath on the other side? Start by just letting your face follow your shoulder around on the rotation, but don't actually take the breath. See where your head ends up in the water when you do that - deep under the surface perhaps? You'll soon find out if you are moving up and down in the water a lot or rotating on a straight horizontal axis.
After doing that for a while try turning your neck slightly, again without actually taking a breath, but as if you were about to.
When you've got the hang of rotating smoothly with your face just enough above the water, try taking some small bites of air on the problem side.