I'm at NAPS myself this year and I wondered that too when I was coming. It turned out that in 3 months there's a really good chance that you'll connect well with at least one teacher here. As long as you show them you're trying and care. Also, for a lot of people if you can't get a teachers rec. then your Company officer or Senior Enlisted Leader will gladly write you a rec.
For the Cadet question, Coast Guard Cadet Candidates here don't need to get nominations to go to USCGA. USNA Midshipman Candidates should be just about guaranteed as long as you don't do horribly academically, pass your PRT, and don't get in major trouble. If you don't get a nomination there are a ton of secretary of the navy noms for that reason