Good day, I first noticed a difficulty in breathing when I struggled on walking up a mountain once. Someone advised me to stop breathing through my mouth half way up and it helped but the reason I was breathing through my mouth in the first place was a mystery.
I often suffered with my breathing slightly before that and afterward it was difficult and I used to yawn a lot in order to correct what seemed to me like pressure build up in my head and if I did not yawn properly I would get a throbbing headache and my breathing would be worse. I am not studying or in a stressful job now so it has improved. However when I read what you wrote about only breathing through one nostril, I have not experienced this symptom, so it is important to find out if that is not related to something else in the sinus area like a departed septum.
Oh on that Sjögren's syndrome, no not tested but suspected. With general dry skin are the other signs puffy eyes and dry mouth and if so should both be present to get tested?