I had a similar beaver issue this year. They turned a seasonal wetland into an awesome duck pond last fall. I thought this was great since I wanted a duck pond here anyway. The problem is they kept making the dam higher and higher, flooding things way, way beyond what I wanted. Neighbors were complaining because the water was backing up nearly to the road and causing drainage problems. I tried cutting the dam and placing PVC pipes under the dam to regulate the water height and the beavers solved that problem by building a second dam below the first.
I ended up shooting them with a shotgun. 00 buck worked well. They are pretty active at first and last light and you can kill them all and then do what you want with the dam. I left my dam and cut a channel in the adjacent shoreline to regulate the water depth. Hopefully that allows the water to stay at the perfect height.