Dry mouth is a very common problem for us druggies
Many on OBT suggested Biotene (pretty sure of that name) toothpaste and mouthwash. It is made specifically for that. You might even call your dentist, as dry mouth is the worst for oral problems. It really escalates tooth decay.
I get dry mouth also, but find myself drooling out of the side of my mouth that is on pillow. Drives me batty. I've never drooled before. The worst for me is the morning feel and taste. Sometimes when I don't sleep well, I get up and brush my mouth in middle of the night.
Don't let dry mouth go!
I have no idea how you live in that climate. I'd be dry in a lot more places than my mouth