Big River is still on the old priority seating system You cant book through Disney Dining, but you can reserve directly with the restaurant. I called in Feb to book for Nov. They told me to call back in Aug for Nov. She didnt give me an exact date to call and sounded a little new. I got antsy and checked a couple of weeks ago (June) and they took my ressie.
The ESPN Club does not take ADR's, so they only accept walk-ins. As for Big River Grille and Brewing Works everything I have read about them says you can make ADR's there, however I have no idea if you can make them threw WDW-Dining or by calling them directly.
You have to call Big River Grille directly. ESPN Club doesn't take reservations. I can't imagine that on a football Sunday if they did. Everyone would have a 1:00 seating time and they wouldn't be able to book anything after that.