So Wolfbloods transform under the full moon, whether they want to or not. It is debatable whether this is really possible - since it's not the moon's light that turns them (cause it's basically just reflected sunlight, meaning they'd also transform during the day). The spiritual energy the moon gives, perhaps? Wiccans and spiritual believers say the moon radiates some type of (positive) energy towards the Erath, it coming with the light. If this is what affects the Wolfbloods' transformations, then it could be plausible that the said energy would make them transform.
So why on the full moon and not first/last quarter? The Wolfbloods would probably need a minimum amount of energy to have the moon trigger a transformation. And it would depend on how strong the pull of the moon is for the individual (which Daniel explains in se1 ep1). It depends per person on how much the moon influences their transformations and behavior. For ones who have a very strong connection with the moon would probably transform even on the day before and after the full moon, as the moon is still 99% full on those nights. They won't need as much "Lunar energy" as Wolfbloods who's connection with the moon is less strong.
Going off this theory, it thus also depends how much the new moon affects the Wolfbloods. Making the wolf-gene disappear in their DNA for one day makes no sense, but they could definitely feel tired and unable to transform during that day, for there's no moon and no energy to make them transform.
Another theory is that the full moon transformation could be a placebo effect. The Wolfbloods merely believe they will transform during that night, strongly being convinced that the moon is making them transform. So these beliefs are so strong that the body will give in to these beliefs and expectations, and thus they will transform on the full moon. So in short, if they believe the full moon makes them transform, it will. If they believe the new moon makes them tired, it will. It is the same as with people who try to turn into Wolfbloods or any other non-human creature by doing spells, rituals or whatever - because they believe it will turn them, they will get the "symptoms" of which they got told were the signs that the spell worked. The spell itself didn't work, the person hasn't actually turned into something else, but by believing they have they trick their own mind. Think about it, if becoming a Wolfblood/werewolf/whatever was so easy you could just do a spell or ritual to achieve your goal, this world would count thousands more non-human creatures, if not millions more.
But as Maddy explains to Shannon, you can't become something you weren't born into.