Took me a moment to absorb that layout, and all I can say is it's ambitious and awesome!
As well as drainage, it would be a good idea to consider what happens to 7/7 water sitting on top of a raising bridge when it opens.
I think any water that's in the space where it becomes a wall will be destroyed, the rest...might well fall through below. I'm not entirely sure. Have you factored this into your plans?
What you do is move the dam bridges out a few tiles, giving you room to have a couple of drainage shafts dug into the bottom of the river bed in between them that head down to the caverns, with grates at the top and lever activated hatches just underneath them or something like that. So once you pop up both dams, you just open the drains and it drains out the remaining water. Or you could have your drains in your entry hall itself, if you don't mind it getting a little wet from time to time.
Alternatively, rather than digging your entrance tunnel into the side of the river wall, you could dig it down into the river bed, and cover it with a lowered bridge (which acts as a false river bed) that will be raised once the dam bridges are raised to reveal the ramp/stair way down into the dark.