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Odd behavior can indeed be perplexing but most of the time it has basis in something that has happened in the past. Since he is a rescue we will not know what set him off but it may have been lots of things. Perhaps in his previous home, he was rewarded with special toy the happened to be given to him after he saw the full moon, perhaps he just noticed it and was unnerved by it- and is trying to warn you of danger, perhaps something more traumatic happened when he noticed the moon previously and he now associates it with a bad experience. If this is bothersome to you, I recommend keeping him in during a full moon so that he cannot see it. You may also try using something to reduce anxiety such as
DAP which stands for dog appeasing pheromone http://www.dogappeasingpheromone.com. This is a product that can be used during times of stress or anxiety to help calm a dog and make him feel more secure in his surroundings. Another one I like for short term issues is called Rescue Remedy and information can be found here http://www.bachflower.com/rescue-remedy-information/
Wearing a thunder shirt can also help him feel secure as well and can be used during times of stress or situational anxiety http://www.thundershirt.com/?gclid=Cj0KEQjw88q9BRDB5qLcwLXr7_sBEiQAZsGja1olZV4dC76JxhSU-8u-qbzvxZ4L32OViCJZQWNgVVcaAk7p8P8HAQ
You can also talk to your vet about drugs as well if you prefer, however these tend to have side effects and may be more extreme for your goals. You may also decide that this is just one of his quirks and makes him unique. Best of luck to you. Have I been able to address your concerns with your dog today?