I'm fairly certain they just cut you a check for an approved budget. I know you do need research clearances for wherever you go, so you may run into trouble if you're already there. I'm sure you wouldn't be the first F-H recipient to begin a grant while you were already abroad though. I would guess a university admin would know more and could better advise.
All three are excellent options, good names and good schools. For me, it would come down to cost: which school will cost you less to complete the degree?
Our grants administrator has said she has not heard anything from Fulbright-Hays. It seems silly to move up the application deadline but not send out notices until the same time as last year. I bet we know in two weeks. Anyone else heard anything?