And from what I understand, a full moon to earth is when it is on the opposite side of the planet from the sun. A dark moon is on the same side of the planet as the sun therefore we see no reflection. With that, it could be said that the larger gravitational pull is experienced at the new moon when it AND the sun are on the same side, and the lesser gravitational pull would be at the full moon when it is on the opposite side of the planet negating or balancing out some of the sun's direct effect.
Does anyone complain about wierdness around a new moon? No, not ever. Because they dont see a bright round shining orb at night. Do the prison populations around the country go crazy when its dark out at night? No they dont.
And to be realistic, phases of the moon have always referred to the shape and amount of perceived light reflected, they have never referred to the actual position of it. So what your references are probably trying to cite here is the actual location of the moon in its orbit with regard to the Earth and sun and not what it looks like. And as such, a new moon would have the most effect on this so-called "gravitational effect" that animals experience and the full moon would actually lessen it.
So for argument sake, perhaps the full moon is affecting things because it is the peak of the least amount of pull of the sun on your brain.
Your computer monitor has more gravitational affect on your brain than the moon does and so does the car, the house, the chair and the planet that you are standing on. You need to actually be studying physics to understand gravity beyond what they tell you on Sunday morning cartoons.