So sorry to hear that. That sucks that your college won’t offer you counseling (same thing happened to me, unfortunately). I dealt with a somewhat similar situation
Not saying you’re not working hard, because you clearly are - but if this works for you, maybe it could potentially be helpful to think about taking on another job while you think this over, even if it’s just folding clothes or making coffee or something. (Maybe you don’t have time for that, though. You know your situation best, of course, and I won’t presume to tell you what to do).
Idk if this makes you feel any better, but I applied for probably 500 jobs after graduating and only got one interview. Even places I thought I would get in, like Starbucks and CVS, didn’t take me. I probably would have gotten hired faster if I’d been willing to move, but I had to stay near DC for family reasons. Most of my friends moved to different cities or even states after graduation, and they all got hired faster than me (not sure how wide a net you’re willing/able to cast, but something to think about).
Eventually I started working at Dunkin Donuts for about four months to pay the bills before I finally got a job in my field. I actually really enjoyed it. Some people judge me for working there with a college degree, but I’m proud of my time there. I learned a lot of useful things like people skills, negotiating with difficult customers, working with different management styles, etc. It made me a more well-rounded person. (If you do decide to go that route for awhile, you can (I think) leave something like that off your resume if you want, or you can leave it on; if you sell it well, employers should appreciate that you have a variety of work experience.)
In re the recommendation letters: Can you get rec letters from people at your internship/s? If not, could you take some more classes at your old college? If you want to get into the Navy’s NPO program, it sounds like you just need those two classes (did I understand that right?) and you could probably take those at your old college, or, if not, at a community college, and take that opportunity to get to know professors and maybe make some connections.
How long is the NPO program versus the engineering degree? Which pays better? Which would you like more? Does the Navy program involve going in subs? If so, something to consider is that some people are fine with working in submarines, but a lot of people really hate it. I hear it can be extremely stressful.
Hope this helps! Best of luck to you. This is a stressful time in life with a lot to consider. I think things will probably become clearer as time goes on. Best wishes!