Getting in right out of school is not going to happen. You will need to put in a few years first. I applied right out of school and was told that my class rank and whatnot was higher than everyone else's, but I didn't even get selected as an alternate. Historically, there's around 10 people applying for a single spot. If and when you do get accepted by the Navy, it's still up to you to apply and Match to a civilian program. I heard a story from a few years ago about someone who was selected by the Navy and only applied to something like 3 programs. They didn't Match and that was the end of the road for them. As a career pediatric dentist in the Navy, I was told by the specialty leader that you should expect to spend 80-90% of your career overseas. There are few billets stateside.
My $0.02, do your payback and pursue it on your own. Like
@averageasian said, you can take advantage of the GI Bill to help fund it afterwards. This is my plan. Best of luck to you!
Big Hoss