I can't speak for the guys in the eastern USA, but IMO here in the Midwest (specifically corn country) hunters are limited on their abilities to hunt entire "drainages" due to the amount of private property there is. Additionally, there isn't a lot of elevation change that visually delineates one drainage over the other.
Speaking from my experience, since we don't hunt a "drainage" we typically hunt smaller properties, and use the term "draw" to describe a small geographical depression within the property to convey where we were hunting. Here is a screen shot of an area that I hunt. In the terms of western parlance, I'd argue that the red line would be a drainage, I'd call that the creek. The yellow lines would be the draws that'd we hunt.

Speaking from my experience, since we don't hunt a "drainage" we typically hunt smaller properties, and use the term "draw" to describe a small geographical depression within the property to convey where we were hunting. Here is a screen shot of an area that I hunt. In the terms of western parlance, I'd argue that the red line would be a drainage, I'd call that the creek. The yellow lines would be the draws that'd we hunt.
