Hey habbaix -- a bit of a tricky situation - let's see what we can do.
First of all, you said you want to go for the January "intakes" - you mean deadlines? Stern has a January start for its P/T program, is that what you're targeting? Just want to make sure we get the right picture.
The 660 is going to be tough for any top school -- but even more so because you're young. We have seen Stern turn down younger candidates with lower GMAT scores. LBS doesn't care about the GMAT score quite as much as many US schools, but LBS does also like ~5 yrs work exp. So the combination may make the schools say, "What's the rush?" In other words, they may turn you down because they feel you are young, you've got time, you should work on strengthening the profile and then they'll consider you.
Of course, very few schools will actually TELL you that when they reject you (some schools do offer feedback if you make it to the waitlist before being rejected but not many do this).
A 660 isn't a bad score - in fact, it's darn difficult to get a score that high! But the quant side may make them nervous, as you have deduced. And, your background doesn't appear to have much (any?) heavy quant stuff at all, certainly not your undergrad.
That undergrad GPA is nice though - the schools will like that! And yay Fulbright! Another plus.
The basics of your background look totally relevant and you seem set up strong... but that GMAT may be a cause for concern, unfortunately. We hate to say it but we do think another shot at the test will help, at least if we're talking these powerhouse schools.
One strategy you might consider, this is risky but it could work: Focus on the NYU essays now, get those built up to be super-duper awesome -- it will take you more time than you expect, believe us on this -- and then see where you're at in terms of timing. Yes, you need to wait a month before you can schedule another test anyway, but you also will want to carve out room to do some good prep for the next attempt, to be sure you get a good bump.
So you finish the essays, and you dive back into studying, and you schedule your test date ... in January. Like, towards the middle or end of the month, whatever you feel will give you adequate time for another round of prep.
And you submit your apps for the January deadlines, and you tell the schools you're retesting on January X.
They *might* hold off on evaluating your candidacy until they get your new score. They do not have to do this, and they may end up assessing on the original 660, or if those essays really are as spectacular as we're sure you'll be making them

they might put you on the Rd 2 backburner... and then when you're fabu 700 comes in they will have an easy decision.
You don't want to apply in the last round, but there may be some different possible strategies to play with for the January round. LBS has a Rd 3 in March that is an option too - and wow is this a long email.
Hopefully this gives you some ideas to consider! Lots to think about, no one-size-fits-all response unfortunately...
Let us know if we can (try to!) help further!!
EssaySnark