I can clear a few things up.
First, the CASH program is not a program to allow someone to finish their degree. It is section 3U of the Enlisted Recruiting Manual, and its purpose is to provide those kids who enlist in the "advanced programs" (limited to Nuclear Field, SECF, MT, CTI, AECF, EW or STG ratings), facing up to a year of time in the delayed entry program, something to occupy their time while they wait.
If someone meets the program requirements, and they already have 3 years of school, then it is possible for them to finish their degree while in CASH, but that is not the typical case. The BDCP is a better option for them.
While in the CASH program, the kid is processed onto active duty, including getting an ID card, full medical/dental care, Base Pay, Basic Allowance for Housing, and Basic Allowance for Subsistance, or about $2000 a month for an E1. That's not too bad if you're 18, living at home and spending $500 a semester at the local community college. They're also eligible for promotion to E2 or E3 like anyone else in the DEP.
The applicant must be in school full time (at least 12 credit hours), and continue to meet all elegibility requirements for their Navy job. I, as the Advanced Programs Coordinator, get to approve the classes the applicant wants to take. The basic idea is to allow the kid a chance to get some college credits, while the Navy gets a better educated sailor, and everybody wins. The maximum time allowable in CASH is one year.
As for officer programs from the enlisted community, the best shot is from NNPS. To meet the requirements to be an enlisted Navy Nuke satisfies most of the officer selection requirements. My current CO used to be the XO of NNPTC, and has confirmed that of the kids who meet the officer selection criteria, 74% of them get picked up. NR actively recruited me while I was on my last sea tour, and there is still the Navy's BDCP (Baccalaureate Degree Completion Program) for those who have completed 30 semester or 45 quarter hours already. Once in the officer world, the carreer path options do include non-nuke positions, but being a nuke is the typical path. A friend of mine, another ETC(SS) who I went to school with, went into the Supply Corps, but it was by choice, not by compromise.
Even someone in the CASH program, if they subsequently meet the officer selection criteria, can get picked up. In my 18 months as the Advanced Programs guy, I've seen it happen twice (the only 2 times someone has applied for it). The individual is given an administrative discharge from the enlisted community and becomes an officer candidate.
The best thing would be to speak to your local Advanced Programs Coordinator about BDCP, ROTC, and Enlisted to Officer program (STA-21) options. Get the details, compare programs side-by-side, and make the best choice for yourself.
If you'd like info on who your point of contact is for your region of the country, I can provide it.
The more information you know, the better decision you can make.