To List, or Not

Definitely use your current job on your resume. It's easier to find a job if you already have a job for some reason. Also not listing your current employer on an application would raise some red flags, after all if you don't admit you are working there a prospective employer would probably think there is a good reason why and that would not be good for your future prospects with that hospital.

Don't jump the gun here. You are worrying about something that hasn't happened yet. Go through the application process at the place you want to work and if you are lucky enough to be hired don't feel bad about putting in your notice at your current job. I wish you good luck, hopefully you will get the job you really want.

Go through the application process, and include all relevant experience (this means your current job). You might not even get an interview, so at this point it seems premature to plan what to say.

It's very common in most fields (not just nursing) to want to keep your current employer in the dark until you've nailed down the next job. Most applications have a box you can tick asking if they may contact your current employer. If you don't want your current employer knowing about your job hunt, tick the no box. Usually, a potential employer is not going to start contacting references until they are pretty sure they want to hire you. So I can't see how listing a current job is going to hurt you in any way. And by listing your current job, you signal that you are a desirable employee because someone else hired you (and has begun to train you).