My nose has a bed in it about halfway down (from what cause I have no idea but it happened when I was very young) and I haven't been able to breathe out of both sides of my head at the same time at night since the 70's. But it sounds like you have a much worse issue. Regardless, breathing problems when trying to sleep are the fucking worst fucking pain in the arse for me. Once I get really tired I start dozing off regardless but then I immediately get the sensation I'm smothering because I have a strong aversion to sleeping with my mouth open as well. So I wake up more than a little startled. I usually resort to trying to sleep sitting up when it gets like that.