My primary concern is that I don't officially have my degree yet, as the school expects to be paid. Is it likely that they will pay my tuition in order to recruit me?
You should read the stickies on this forum, which will detail the various (or singular) commissioning sources available to you depending on where you are in your education.
NUPOC is a form of a Bacchelorate Degree Completion Program. They will not pay off your loans for you, but you will be paid as an E-6 from the time you swear in to the time you commission. You will also receive BAH for your hometown and enlisted BAS. You will also receive a $15,000 bonus (which comes to about $10.5k after taxes) upon signing up. You will have no military obligations while attending college other than to get > 3.0 average. The downside, similar to the pilot BDCP program, is that you will owe two years enlisted time as an undesignated seaman if you DOR from the program after receiving your pay.
I know nothing of the intel community other than it's extremely difficult to get into. If you are the ideal nuke candidate (strong engineering background), you are probably not an ideal intel candidate (strong history/poly sci background), and vice versa.
The Navy doesn't offer to "pay off loans" for officers. We make a lot more in base-pay than our enlisted counterparts for comparable time in service, and part of the higher pay comes the responsibility of taking care of your own bills.