In practice situations I was performing much better. I just think my nerves got the best of me. I just don't know what's worse - applying with my current score or applying round 3 - darn GMAT!
Your current score of 620 is at the lower end of your target schools' GMAT range.
Based on GMAT score alone (ignoring all other parts of your application), Columbia is already out of question. Many early decision Columbia applicants got dinged with GMAt scores in 700 range (not to mention impressive profile).
NYU is very competitive this year. They give out few interview invites as well. If you look at spring 2009 NYU thread, you can get a sense of how competitive even their PT program is this year.
Yale just started releasing admits right before the x mas holiday. Although Yale isn't considered a heavy weight such as Columbia or NYU, I am sure applicants are competitive this year.
Therefore, unless your application/profile is very unique (such as you are an Olympic Gold Medalist or something), it is very difficult to even get interview invites with GMAT score of 620 to the schools mentioned above. (again - only looking at GMAT score only)
R3 option is also unappealing. R3 probably gives an applicant lowest chance of getting admit for top schools.
However, if you are indeed getting scores above 620 (such as 680 to 700 range) in your practice CATs frequently, then you should schedule another GMAT exam as soon as possible and apply R2.
Otherwise you will have to apply R3, or apply R2 with 620 or consider lower tier schools.