@cholash I use to wake up with a burning sensation on my tongue and dry mouth as well. Not sure if it was due to my, then not now, sleeping on my back. I was told/learned that I slept and breathed through my mouth at night while sleeping on my back. I changed the way I sleep.' I have done three things since then.
1. Sleep on the wedge pillow on my left side and
2. I use a small piece of 3M Nexcare tape to keep my mouth closed at night.
3. I stick mostly to the acid reflux diet just to error on the side of caution. NJH said, after testing me for acid reflux, I don't have it.
I no longer have dry mouth or the burning sensation on my tongue. I think the burning may have been to acid, I don't know that for sure but to me it sounds and felt like it could have been acid coming up at night.
This was my experience, don't know if it might help you but thought I'd mention it.

Also, are you able to ......and do you do any type of percussion or postural drainage.
I find that if I have been keeping myself hydrated and do all other aspects needed to clear the mucus I can get just as much mucus up without doing the saline.
So I am trying to do one day of saline and one day without. On both days , with and without the saline, I do all other necessary parts of our BE routine for clearing the mucus.
For me that means..... light exercise first thing, then sipping warm filtered water before the saline, or the day without the saline, and then for both days ....percussion on my front and back at the lung areas, postural drainage, autogenic air way clearance, and huff coughing. I do several sessions, or call them sets, of the four things mentioned.
Hope this may give you some worthwhile information....and maybe something to discuss with the doctor's since the doctors(s) know you and your medical circumstances.
Barbara