Man, you REALLY need to stop worrying. Let me break the entire thing down for you so you stop stressing yourself and focus on enjoying your last year of freedom.
GPA: 4.0 puts you at above 75% for every single school.
LSAT: 170 puts you at the median or above it for every school Stanford and below, with the exceptions of Columbia, NYU.
For the bottom half of the T-14, your numbers alone are enough to admit you with money, as you're above the GPA 75% and close to the LSAT 75% in most of them. There's no way in hell you won't get into a T-14, and you need to just accept that. LSP may be a little off, but it's not that inaccurate - check LSN for confirmation of this by searching people with numbers similar to yours.
Now, different TLSers will give you different opinions about the value of softs, but here's one thing that's certain: a Fulbright is most certainly an immense asset. What it represents is excellence in academic achievement and an ability to perform at a high level and under pressure, two traits that T-14s care deeply about. Sports, work experience, leadership - none of those are particularly persuasive in an application. But, since you have a good set of numbers that make you a qualified applicant at almost all if not all the T-14 (again, HYS are indeed a bit of a stretch but they're not your goals) the Fulbright can go a long way in helping you to get admitted. The Ivy background certainly doesn't hurt either in that regard. So, here are my predictions for each of the T-14. Hopefully this will give you a little confidence that you WILL go to a top law school and that you won't be living off tuna and ramen for the next 30 years of your life.
Yale - very weak shot, but they have accepted those with weaker numbers but interesting backgrounds. Its academic orientation means it will value the Fulbright a little higher than most. Still worth the app fee if you would ever consider going there.
Harvard - numbers give you a weak shot (less than 50%), but again, certainly worth the app fee as your academic background could give you a small boost here. LSP gives you a "strong consider", and a friend of mine got in to HLS with similar stats to yours a couple of years ago.
Stanford - better chance than the two above. Excellent GPA will be taken seriously here, and the Fulbright will supplement that. Your LSAT is at their median, so you stand quite a decent chance. Palo Alto may be far for you, but the weather's damn beautiful and it'd be my dream to spend 3 years there.
Columbia - lower LSAT does put you at a disadvantage, GPA helps but who knows? Hopefully they get impressed by your WE/academic record or are in a good mood the day they read your application. Cross your fingers, there's certainly a good chance, but don't bet all your cards on it. They certainly are LSAT whores, and the GPA doesn't mean much here.
Chicago - good chance. You're at the LSAT median. Money is pretty unlikely as it's mostly based on the LSAT.
NYU - better chance than Columbia; again, cross your fingers and hope. Between the two of them there's a good chance you'll get one, but it's by no means a certainty. What type of law are you interested in? If public interest or international law, you could be a candidate for a big scholarship (though not a big chance of that).
MVPB - in, for sure. I'd hope with a little bit of money, but you're certainly in.
Duke, NW, Cornell, Georgetown - 100% in to all of these. You were wondering about Cornell - the average GPA is around 3.67 and LSAT around 167/168. Do you really think they'd reject an Ivy Fulbright 4.0/170? Good chance at money in any of these, perhaps a lot but I'm not an expert on that. Check LSN for past recipients of money from these schools.
In short: you have absolutely nothing to worry about. A diversity statement will boost you a tiny, tiny bit, but the schools will already see that you have something to add to their class (academic excellence). Relax, do some research on where you'd wanna go if you don't get into Columbia or NYU (pick a favorite or two from the lower T-14, and give some serious thought if you'd attend HYS if you got in. I really do think you stand a good shot at Stanford, and honestly, coming from a damn cold city I'd love to spend the next three years of my life living in that paradise. Anyways, make a couple of scenarios and start working on personal statements. Write a kickass one and you could very well be attending NYU or Columbia next fall. Use the next couple of months wisely.
Last note: in regards to seeing someone for help, as you really do want to kick ass on your PS to boost your C and N chances, contact the academic advising at your undergrad (or just walk into Columbia and pretend you're a current student). The advisers at these elite schools should have a good idea of how to help students prepare apps to get into other elite schools. HYS and other T-14s have high proportions of Ivy undergrads in their law schools.
Best of luck! Hope this helps.