Just to answer the first part here: it will not lower your chances. I had Harvard picked as No.1 and Johns Hopkins as No.2. And I got into my No.2 which is the best school in the world for what I like to learn about. I am not an "A" grader, I am not some super Einstein woman, so go for it! Fulbright is the opportunity for you to see and learn subjects that are in most cases non-existent in your homeland. So, any school, and trust me, they do pick well, is a bonus. So many opportunities, so many open doors, and as I said on my interview : every school is a bonus, because I have none of that back home.
Good luck!