I don't want to paint you a bleak picture, but I'll just tell you my experience...
We relocated to Utah last summer. 6 months prior to the move, I started looking for a job. I probably sent out well over 150 resumes in that time, applying for jobs at all the facilities and hospitals you mentioned. In that time, I got 2 interviews and lost out both positions to someone with experience. (I have no experience either, at least not nursing experience.) I'll be honest with you, if you can relocate or don't mind doing some sort of commute, that may be your only option. I actually went and talked with someone at UofU HR and asked her about the outlook for hiring new grads. She said that for every RN position that they have an opening for, she usually gets about a hundred resumes for it
And, just to add, it isn't my interview. Meaning, I'm not going in and bombing the interview. In my first career, I got every job I ever interviewed for. Even when we were wrapping up nursing school, they made us go through a mock interview and I was the only one in the class to get a perfect score according to one of our nursing faculty. So, not to toot my own horn, but just to point out that it isn't my interviewing skills. It's my lack of experience. Utah is the worst place for a new graduate nurse UNLESS you networked while doing your clinicals, landed some sort of internship/externship, or the ol' "know somebody." But, if you aren't from here and just recently relocated...I'll just say don't get your hopes up.
Just to add, we are military -- you wouldn't happen to be military, too?