The OP would need to provide evidence that is both accurate and irrefutable regarding any claims of the "instability" of the mother to achieve any hope of being given full custody.
Understand that judges who handle these type of cases hear all sorts of wild stories on how one parent is "better" than the other at raising a child. Unless there is hard evidence to show this, full custody is not going to happen.
The general feeling of the court is that every child needs a stable family situation; that includes a mother AND a father working together (even if they are not living together) to provide a safe and secure environment for the child. In the courts eyes, what the parents "want" in terms of punishing their former bedmates means....absolutely nothing.
The court assumes (and often this is an inaccurate assumption) that parents are, well, adults, fully capable of taking care of their own needs. The child, by comparison, is seen as the helpless one in all of this, requiring the protection of the courts if the parents lack the capacity to work out the details of adequate care by themselves.
Gail