In the United States, admission to graduate school is tied to the ability to offer support. If they can't give you financial support, they don't accept you 99/100 times. If you come up with your own funding, you have a shot at a wide variety of schools, not just schools ranked in the 70's. The problem is most funding agencies have similar criteria as admissions committees, though I admittedly know nothing about the Fulbright process.
If you have the Fulbright, you should be applying to schools ranked 15 and down, I think--having your own funding opens a lot of doors. You should be contacting professors ahead of time though, preferably ones that can use your engineering background. The school needs to know that when your fellowship runs out, there's a professor willing to take you.