Coming from the opposite and having asked exactly the opposite question here, switching between different climates is now very familiar to me. It really depends on the individual as some people just seem to adapt better than others. With time though, most do feel a little better but I personally never got completely used to high humidity places, so perhaps you will not completely manage in very dry places either.
The damp cloth is an excellent trick. I use it even just to slow down very hot air from entering my nose in places here it is too hot for me. Keep doing that while you sleep since you need to recover well while you travel.
Now, if you are staying in a hotel, some places have humidifiers available. There are also wet fans that sprinkle the air with water. Personally these things make me feel much more uncomfortable but many people must like them for those to be popular.
During the day and as you go on with activities, use a non-medicated nasal spray. Saline solution is the most common but there are some with aloe that last longer as moisturizing agent. I recommend not to overdo it, so that you get slowly more used to the climate by progressively breathing drier air.