The two conditions are likely very closely related. Your stuffy nose probably causes you to sleep with your mouth open, so that you get enough air in. The mouth breathing while you sleep causes your mouth to dry out, especially since you live in the dry high desert. As you go through the day, you swallow and moisten your throat, and the dryness seems to improve.
So what to do to help your situation? Allergy medications over the counter were a great first place to start. Unfortunately, they didn't seem to help. Nasal saline irrigation a few times a day is a next great step, as it helps to moisten your nasal cavities and helps things to drain appropriately. If that doesn't work, the next step would be to see your primary care doctor
or an otolaryngologist
(ear-nose-and throat surgeon
) who will be able to discuss these issues with you in more detail, do a physical exam, and perhaps recommend an intra-nasal steroid if they feel that your symptoms will be improved by reducing the inflammation in your nose.
As always, visit a physician sooner if you feel that your symptoms are getting worse, not improving, or you have other concerning symptoms.