I think you're missing the big picture here about what you are obligating yourself for by enlisting in the US Navy.
Your first 24 months you'll be at training commands. You wont be given an opportunity to attend off duty education, because your schooling is your job. You wont be eligible for any kind of education assistance, because you are in training; you aren't permanently attached to the command.
If you only plan on doing a 6 and out tour, your entire remaining 4 years will be either on a ship or a sub. The first year there, your primary focus will be on getting qualified senior-in-rate; meaning qualified all of the requisite watchstations and your final senior level watchstation. Plus continuing training. And maintenance. And cleaning the engineering spaces. And standing duty. And performing general military training.
And hopefully you start to get the picture. Your posts almost sound like you think there is going to be some big spots of free time where you can go and take a couple of college classes here and there and finish a B.S. degree. Not. Going. To. Happen.
If, and that's a big IF, you manage to end up on a carrier going into the yards for overhaul and refueling, you might get an opportunity to attend some classes on a somewhat routine basis.
Enlisting in the military means you are volunteering to serve your country; you become the property of the US government and they aren't paying you to get a degree. They are paying you to do what they tell you, when they tell you.
This isn't meant as a rain on your parade response; these are realities of what being an enlisted member of the US Navy is about. Being underway, working long hours, putting up with people you don't like and can't escape, because you all live within 75 feet of each other (at least on a carrier the berthing was that big), with your only private space being a rack literally the size of a coffin.
Engineering rates, both nuke and conventional, are the first ones on board for an underway and the last ones to leave when you pull back in.
The recruiter isn't lying to you about the potential college credits; some colleges will give you some credits. Excelsior and Thomas Edison State College have degree programs tailored for Navy nukes. However, these are Engineering Technology degrees, not Engineering Degrees. There is a definitive difference between the two and you can't substitute a Eng Tech degree in a lot of instances.
Do some research on your own, away from this site. Google "Navy College" and "navy+ degree" and see what you come up with.
6 and out makes you eligible for an entry level position as an Auxiliary/Non-Licensed Operator at a civilian plant. Period. Along with every other 6 and outer that is going to be competing with you for that job.
Lastly, if you want to be a doctor (I assume this from your original post about wanting to go pre-med after the Navy), being an enlisted Nuke isn't going to do anything for you, other than take away 6 years of your life. It's not going to put you in a position to finish a Bachelor's degree that will get you into med-school. Sorry to sound harsh, but this is reality.
Good luck.