A .22 is not a good idea. I've shot them before and still had to kill them later, after a good hit. You need more gun, if you're going to shoot them. They're tough.
I trapped them for several years. I used leg-holds because at the time I couldn't afford kill traps. They key is to set the leg-hold traps near deep water, and use a cable with an L-shaped connector with a hole drilled in it that allows the chain on the trap to go down, but the L keeps them from going back up. The cable has to have a good weight on the end to hold the beaver down and drown it when it dives into the water to get away.
Maybe now you know why the kill traps are considered better. More humane. I quit because I wasn't comfortable with the method. If you don't drown them they will chew off the leg or foot to get free.
Dan Quillian introduce his son D. D. to bowhunting by setting him up on a beaver pond to shoot a beaver when they came out at dusk. It was D. D.'s first bow kill. Break the dam and they will come.