I feel for you. I live in the country too and I know what it's like driving around dodging rabbits, cats, game birds, deer etc. Sometimes there is just nothing you can do to avoid the collision.

Most people here don't even stop to move an animal that they have hit off the road, even if it is clearly someone's pet and not a wild animal, My husband seems to regularly leap out the car to lift animals off the road.

Please, don't blame yourself, you did everything you were able to do.