Honestly, with your background I'd recommend joining as am HM (Hospital Corpsman). They have specific NECs for some of the things you're already qualified to do and you'd be able to continue working in the medical field. They also have a little more variety in where they can be stationed.
As far as education, you are correct, you will get very few credits to help get an Engineering degree, most colleges that cater to the navy nuclear power program offer only engineering technology degrees. I'm not sure what all the requirements are for an Engineering degree, but, theoretically, you could still get one, it would just take longer to accomplish. As an alternative, you could apply for one of the many officer programs, either before you join or after enlisting, and have the navy pay for your education.