I'm not sure what your question is.
The Fulbright doesn't cover a masters degree. It funds different things like Independent research (while engaging in cultural exchange) or Language exchange (while engaging in cultural exchange). You would do your masters before or after the program, not during.
Additionally, apply through your old school. You'll have a wealth of resources you wouldn't have otherwise... and it wouldn't make sense for them to not allow you to (unless they have an overwhelming number of applicants) as it looks good for any school to send someone into the Fulbright program (I graduated in May 2011, I applied (through my school) after August of 2012.