There have been cases that both the tailed beasts and their jinchūriki are affected to a certain degree during a full moon:
According to Gaara, Shukaku's blood-lust is especially strong during a full moon.[26]
In the anime, Blue B, during his time as Gyūki's jinchūriki, noted that he would have trouble sleeping during the nights of the full moon, because anytime he would stare at it, it would reflect the loneliness in his heart. Gyūki would take advantage of those moments of weakness and try to control over Blue B's body. Blue B was also shown to take sleeping pills to help with this side-effect