Well, first of all, I'm sorry to get to this so late, as with less than a month left it will likely be difficult to gain much from any sort of exercises.
That being said, the best type of exercise for adjustment of this sort would be running, ideally with a sauna visit directly afterwards to simulate the dry environment. Running serves two purposes here. The first is to strengthen your leg muscles, in an effort to make motion easier for your body. Easing the effort of actions like walking will help with the transition, as if your body is stressed you will be more likely to experience the noted symptoms. The second purpose of running is to induce fatigue, so as to make the sauna trip more effective. If you can get adjusted to a very dry, hot sauna, while fatigued, then the new country should feel comfortable in comparison, which will also help with the symptoms.
While exercising in this manner may help, you will still experience these symptoms to some degree after the move. My recommendation unrelated to exercise is to find a nasal spray and moisturizer you like, and then make use a habit. With such a short time until the move, these items will probably be your most effective tools.