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He seems quite hard-wired. We shall see. He may turn out to be my foster dog rather than my adopted dog. I certainly won't dump him off at a shelter, but after a bit more time if he doesn't get better I will be looking for a critter-free home for him.

That makes sense. You both need to be a good fit for each other. In the meantime, I'm sure he's happy to not be a neglected hunting dog and not be in a shelter.