Welcome to GMAT Club and congrats on the Fullbright and Your Admits!
1. I would not take Babson's offer and would even question about going there even if it were a full time offer
2. Tulane - it is a well known undergrad program and making it free, makes it pretty low risk. The upside is not great but if you are open to networking, it is the best option.
3. In terms of your goal - is it PM role? These are VERY hard and VERY competitive and require a Top 20 for best chances/accessibility. You can still apply for jobs but it will be a lot tougher to break through unless you can network and get your name out there (you need a plan B really as well)
4. Most people pay full price for their MBA degree. Sure some get scholarships but the Top 10 is going to be $120 on tuition + $60K on living expenses and if you think those numbers are out of your reach, I understand the discomfort. We all have comfort limits.
If you are looking for a PM role you need good strong work experience. Do you have it? If not, probably not worth betting on it as you won't be able to compete really for the best positions and jobs.
My suggestion is that if you can't handle the debt, you take a free option or don't get an MBA.
In this situation, you are not losing a whole lot by getting a free MBA from Tulane since you don't seem to value your time all that much and don't feel you can make $120K. Seems like a good offer to me.
BB.