Hi, yousefa! You have a really interesting background.
So, to your questions:
1) Do you need a higher GMAT than a 680 if you're applying to HBS, Wharton, Stanford and Columbia? I'd say yes. Basically, you'll want to be as close to the schools' median/mean score (schools report differently) as you can get. (Stanford= 737; HBS=730, etc.) The schools often also publish their full score range or their mid-80% range, and a 680 is either within or dangerously close to the bottom 10% for the schools you are naming. Do they make exceptions for incredible candidates? Yes they do, but it is risky to count on that, and as an "admissions expert," I would advise a retake.
2) Can you take the GRE instead? Sure. All the schools you name accept the GRE.
In fact, if you're scoring below the average, the GRE is probably a better bet since the schools really do seem to focus on trying to keep their GMAT averages up for reporting purposes and the rankings. However, with the rising popularity of the GRE, those scores are not exactly ignored and at least one ranking (US News and World Report) is beginning to include them in the calculations. But if you're more comfortable with the GRE, go right ahead and take that instead. Check out the schools' student profile pages for an idea of where their GRE ranges are - many schools are reporting them now as well.
Best of luck, yousefa! And good luck with your new job-- very exciting!
Julie-Anne Heafey
Senior Consultant
mbaMission