Do you run anything other than 400/800? No 1500 or 5k times?
The very high SAT scores being quoted by others are not needed if you are actually being recruited and the coach is supporting your application. One score over 700 and the other two 650 range will get you in if they want you. A 2100 would be very good. And I know what I'm talking about - personal recent knowledge. But they have to really want you, and your current times are not quite there.
If you are not recruited, however, you will be competing with the general student body. And not even 800 SATs will assure you of getting in. Both schools admit well under 10% of applicants and most are very well qualified. It's pretty brutal.
I know an 800 runner with numbers very similar to yours who did not make it into Princeton - he spoke with the coach but they didn't push his application. But he was accepted by another very well known, prestigious Ivy League school that had more need for a runner of his abilities.
Stanford currently is without a distance coach or head coach. Hopefully they will bring someone on board soon. So recruiting will be on hold most likely for the next several weeks. Just so you know.