Some good advice here. My friend has a NSH (half sdlbd and half arab) who also runs with her nose in the air and hollowed out. It is such a rough ride, and harder than heck to find a saddle taht fits a hoollowed out back like that.
I agree ; let her trot out freely (you post the trot or 2pt it) for a good long time without asking anything from the bit. In fact , push her even faster than she wants to go for short periods of time. Like a nudge to just get her to speed you, then let her coast down to her natural rythm but trot, trot trot! After she has slowed down you start to take up some contact and again ask her to go faster. If she raises her head just maintain soft contact and lightly tickle the inside rein to invite her to drop her nose, If she does, then give a HUGE release and let her come to a stop if she wishes. Walk on a long rein with pats, and start up again. You let her trot her way first, then it's slowly uptake contact and ask for a "give", you reward, relax and start agian. As you go along the "she trots her way" become shorter and the "here's some contact" gets longer. If you get something like 5 good gives, you are done for the day. Don't worry, she will remember .
Also, do you know how to "show her the way to the ground"? It's a cowboy thing that they teach the horse to reach down to the ground while they are mounted at a standstill. You kind of very gently take up minute contact and keep your hands out to the sides, like you are driving a wheelbarrow, and then vibrate one rein and then the other and if the horse reacts , you give your hands forward to any reaction. As you go allong the reaction that you want is for the horse to reach downward and outward, and when she does you follow her. The better you get at this, the better able you will be at giving downward to follow her without losing contact. Initially, you might have to give big, and have aloop in the rein , and might have to start from the beginnnig over and over. I also say, "Down!" when I do this. This is an excellent skill to teach a horse becasue when their head is downward they have a physical reaction of relaxation. IT's hardwired in.
YOu can practice showing her the way to the ground while standing or if she reacts by walking off, just ignore it and stay focussed on her reaching forward and downward and only then does she get the reward of release.