Apparently there are rumors brewing that the dog execution story didn't go down exactly as Noem said it did. Basically, she was in a fit of rage and when she gets like that and there are firearms nearby, something is gonna get shot. And because killing the dog didn't satisfy her blood lust, she also shot the family goat for good measure, just because.
The reason she put the story in the book was to get ahead of it before some other source leaked it. So she wrote a self-serving version of events to promote a narrative. Since the book release, however, she's augmented the story with more bullshit about the dog threatening her children in order to justify shooting it. But there were witnesses and she won't keep the lid on it for long.