Thanks for reaching out. First, let me say that your choices are not mutually exclusive, meaning that you can apply to Texas/UNC/Kelley for round two and then NYU/Yale/Duke for round three if you want to.
What is interesting is that the timing of both approaches makes more sense. It's probably not going to be productive to apply to Yale and NYU with your 620 (Duke is more of a possibility, but barely), so waiting to get the right score makes sense - and these aren't schools that are going to wonder about your interest if you wait. Know that there are going to be fewer spots in the class for Rd 3, but it's worth it to boost your score.
As for the other slate of schools, I think you are going to be an interesting enough candidate that you can be competitive now ... in fact, waiting until Rd 3 is going to introduce questions of interest. They might think you are just throwing safety school darts at the end of the process. It's going to take really well crafted essays to showcase your intellect and other key themes, while also hammering fit with these programs, but that's certainly something we can help you with.
My advice is to apply both Rd 2 and Rd 3, and I would consider Tepper instead of UNC and if you get your GMAT up, I would consider Booth over Duke.
PM me if you want to get into this a bit more with an initial consultation.
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