Got the job...

but wont be starting til after feb but HR told me to turn in my notice this week, the job is in the same hospital system. She said all she needed was to talk to my manager for a reference (see if i have any marks, or write ups which i dont.) I was hoping to hear back from HR in regards to if she talk with my current boss, But i havent heard anything. I should call HR and then give the notice. Should i be uneasy about giving the notice this week, it just seems too soon.
Have you signed the official job offer? Some places require 4 weeks notice, but I would make sure I have something in writing from HR asking you to give notice and offering the second job.
Agreed. Verbal offers mean nothing these days. Ensure that you have a written offer, which includes your position, rate of pay, and start date from HR. Do not give notice at your current job unless and until you have such document. Verbal offers are easily retracted.