Cara, I agree, this dog should not be working livestock, but the OP never said she was using the dog for herding activities.
I have ducks, chickens, turkeys, and rabbits, I expect my dogs to not touch them at all. The shelties I have had no problems with this. A couple of verbal corrections to leave it was sufficient for them. The guide dog fosters on the other hand can sometimes being a little more stubborn about it. While I never leave the dogs unsupervised with the livestock, accidents do happen the chickens and turkeys in particular a good at flying over the fence, or they stick their heads out of it. A good hard leash correction cured most of the guide dog pups not to go after them! We did have a boxer that was just to high in prey drive to get him to stop. He was washed from the program by the trainers.
Hanne, just because they are having troubles right now doesn't mean they need to rehome, or that the dog would be happier. GSD's tend to have a harder time readjusting to new homes. Training is the best thing they can do for that dog, especially obedience.