I have gotten the burning dry mouth on several occasions, and it lasted for weeks. I started using dry mouth gum and mouthwash (not sure if it helped).
I still get the dry mouth at night (bad), probably from mouth-breathing.
I have always assumed the mouth issues are from sinus and allergy problems: inflamed mucosa, swollen turbinates, poor air flow.
What kind of sinus exams have you had? Did they do a CT Scan? Endoscopy?