Hey kamenashi! Not "way lower than average" at all! (OK maybe a tiny bit...

but those are AVERAGES!)
You've got a lot going for yourself and we see plenty in this brief sketch that the adcoms may respond well to. You've also named
some of our favoritist schools.
We don't see that many women applying with a chemical engineering degree so right there you are at an advantage, in terms of presenting some diversity to the class. Your grades are totally in range, as is your GMAT score, and this looks like a good time in your career to be going for the MBA.
To correct a few misconceptions:
1. The schools do not actually care about supervisory experience, necessarily - it can be additive but it's totally not mandatory. What's much more important is evidence of leadership and impact (you can see that from the essay questions - Tuck #2 and the Darden question show it clearly). Someone can have great stories of doing great things on the job and not formally be a supervisor or team lead or anything. So don't get too hung up on whether or not you've literally had people reporting to you before. It's all about how you've made contributions and added value (and this can be defined in so many ways!!).
2. Extracurriculars are also not nearly as important as some people believe. Yes they can be additive but we see plenty of strong candidates get in every year without having a boatload of volunteering or whatever in their profile. The schools look at the whole picture. If that's part of the package then great but it's certainly not necessary to get in.
We have a bunch of posts on the EssaySnark blahg about this and other issues all the time so feel free to check them out - this one about
women at Columbia may be interesting as a start, just to get a glimpse of how the schools are trying to recruit more women and what the on-campus experience is list. We've also got a bunch of essay reviews posted from all the schools on your list which can be helpful as a start in figuring out how to position yourself.
On the GMAT specifically: Your current score is below the average at the schools you've named but it's not too low - though if you feel you could do better, you should certainly give it another go. We strongly recommend that you apply this year, however, since you're moving to the 'older' side of the equation, and anyway, it sounds like you're ready and now is the time. So don't delay things till next season just to wrestle with the GMAT (and make sure to apply in the January rounds - Round 2s - anytime later is a major disadvantage).
Phew! Hope all that helps somewhere.... Let us know how things go for you and good luck with all of it, you can totally pull this off!!
EssaySnark
kamenashi
Undergrad: UC Davis (US Top 50)
Major: Chemical Engineering
GPA: 3.24
GMAT: 690 (Q/47, V/37, AWA 6, IR 7)
A/S: 31 Female
Work Background: Engineering in Oil & Gas
I worked at a mid-size EPCm (engineering, procurement, construction management) firm for almost six year as a process design engineer. Besides the usual number crunching, I regularly interfaced with clients and incorporated their needs into the design, and I coordinated the project tasks among various departments to ensure that goals were met at a timely manner.
I wanted to learn more about the oil and gas business, so I decided to work for my current employer, an independent refinery. I have been at my current position for less than three years. In addition to doing everything I did at my previous job, I now deal with vendors and manage contractors for engineering and construction projects. I know the schools like to see people in supervisory roles; while I don't have anybody reporting to me, during constructions, I give out orders to get things done (does this count as a supervisory role?)
In terms of extracurricular activities, unfortunately I don't play competitive sports or volunteer at NGOs. I see this as a disadvantage.
Please evaluate my chances of getting into Tuck, Ross, Duke and Darden. I'm on the fence about applying to Insead and Booth because I think that my profile is way lower than average. Please also let me know if applying to those two schools would be futile. Thank you very much.
Best Regards,
Terry