I had this exact problem when I moved from a big city to a very small city. First of all, there weren't really any of the type of job I had done. Everyone assumed when I found a "good job" that I would leave, or that I would want to take their job (the managers). So you're going to have to put together a strong story as to why you want to change careers and why you always wanted to work for a hospital (university). I would put that story right in your cover letter. No one wants to train someone and have them leave a year later, so they will assume that if you don't tell a compelling story.
If you're interested in working at the hospital, maybe you could sign up to volunteer and that might get you a foot in the door.