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You did very well to protect your chickens. If you don't have a gun, submerge the whole thing in water, make sure it's all submerged and walk away. Come back in a half hour. There is no need to traumatize yourself over what happens when you're gone. It's unnecessary and it's not going to bring comfort to anyone, nor will it make up for the dispatching, or even return the raccoon. There have been a few suggestions as to handling the cage itself and it's good advice. You don't want to touch it's saliva, and you don't want it scratching you.
Getting rid of predators is part of the deal. At least you're not trying to relocate it, or irresponsibly putting your hands on a gun that you don't have experience with. Without the gun (which is the most humane), drowning is the next best thing. Don't think twice about it. Letting it out of it's trap just releases a raccoon who will be difficult to catch later.