I do not believe the age requirement is waiverable. It was not in the past and I do not believe it has changed. But it's always good to verify. Maybe things have changed!
I will be 100% honest. Take the extra year. Yes, USNA is very familiar with your school. Know plenty of USNA grads who attended Exeter. Graduating at 20 is not ideal. Trust me reporting to a ship at 20, it will be noticed by your Sailors. 22 seems extremely young as an officer and can be overwhelming. There are lots of new grads checking into ships right now from 21-25 years old thinking what the heck? It's great you want to be an officer, but as an adult now, don't be in such a rush to adulting. You can never go back and get that time. The youngest person in my class was a grad of your school and he met the age cut off by 1 or 2 days (he was in my Plebe Year squad). Go have some fun, travel, go to a normal college for a year and if USNA is what you want, it will happen. Your scores are top notch and your school has an amazing reputation. You are familiar with USNA, so you know that if its not waiverable, there are no exceptions. If that is the case, you got a year to figure out. Looking back now, many of us would of loved to be in your shoes.