Looks like your score should be good enough for the schools at which you are looking. Plus, a 4th score that is much higher will look more like the outlier than anything else with 3 clustered scores. Focus on your essays.

akuma wrote:GMATPrep1: 670 (47Q/34V)
GMATPrep2: 680 (49Q/33V)
Real GMAT: 680 (44Q/39V)

I am at a loss of words. Quant today was unbelievably hard (I literally wanted to cry). Any suggestions on whether I should retake for 700+? I feel as though my verbal score was pure luck and I'm afraid a retake after a more extensive quant prep would result in a lower verbal score.

I would say it's worth retaking (judging by your scores, you've only taken it once?). Look at it as you can do no worse than the 680 you currently have. Also, depending on what your academic background is, a 44Q score may raise some concerns as some schools like to see that Q score in the 70%ile to get the warm fuzzy that you can handle the quantitative nature of an MBA. So even if you got a 680 again with a higher quant and lower verbal, that could well be a "better" score.

A higher GMAT will also put you in the running for scholarships and helps with recruiting.