Get your self a couple Conibear 330 traps. They kill almost instantly and can be set on land or water. A Water is the best and it keeps most other critters from getting in the trap. I have caught a few turtles (bonus for a pond owner)and once I even caught a 6 lb bass.
It's really easy to trap beavers. No sence in paying some to do it, sounds like you'd enjoy the activity. Take a couple traps, set them, take the easy first beaver shot with your rifle that evening and let the springs do the work from there.
Find the fresh slides where they are going in and out of the water and set them where the top of the square is right at the water surface. A few sticks stuck in the mud on either side of the trap will help direct them through the trap. In the shallow water you can see where they will cut channels. These make great set opportunities as well. There will probably be a mating pair and possibly a couple youngsters you'll have to deal with depending on how long they've been resident. Once you take out a couple beavers the damm is yours to do with as you will.
Dam removal is actually kind of fun and there is no real danger of the "levvy breaking" unless you plan on dynamiting. The dam was built up a fraction of an inch at a time with water behind in and can be removed little by little the same way. I have removed several by hand and have never had a problem. It was built with material that a 60 lb rodent can handle so it's about position not brute strength. Beaver dams don't cave and collapse like earthen dams. You can cut a section and the surrounding sections will hold Use the hrdro force to help remove the dirt and debris. Send it down river to the gulf. If you drain first them You have to remove everything and clean up the mess by hand. A backhoe is handy but not required. If it's only 10' wide you don't have an insurmountable problem. The worst part will be waiting for the water to drop so you can do some more removal. I'm guessing you could have it out in a day, maybe two.
