I'm likely to be separated from the Navy soon, and it's probably going to be OTH. Am I completely screwed?

Go to medical and get tested for sleep apnea, tell them that you are constantly falling asleep. This is critical. It may wind up getting you a medical discharge which is a lot better than an OTH. First go and see legal and see if they can help you out. DO THIS TODAY OR TOMORROW. You do have the right to an attorney at Mast, but you'd need to pay for it. Last resort you can ask for a special court martial, though it is not guarunteed if you are on sea duty.