Don't undersell your GMAT score - a 680 is definitely a good score, although if you're confident that you can achieve higher, there's no reason to not work for that.
What's most important to consider in the month of December is how much time you'll have to complete your applications. While you're correct, you don't want to dilute your effort and focus between too many schools, also keep in mind the high level of competition for the programs you've selected. It would be difficult to tell any one candidate that they will get into any of these programs without looking at the complete application package. Business schools are concerned with your overall capabilities, not just one aspect such as the GMAT or your professional achievements.
That being said, I would recommend you begin your applications now, even while preparing for the GMAT again. Why wait until you have taken it again in December? You should have enough time to get recommendations and transcripts requested, which mainly leaves the essays. Be careful that they are specific to the program, which will take time, but you can use elements from your experience that are common between them. If not all 3, I would definitely recommend at least 2 of your desired programs.
You'll also need to address why you are interested in receiving an MBA from the US when you've gone through business school in India already. How has this prepared you to further your education again? What can you offer to a program, and get back from it, that you haven't already?
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