Is not communicating with one's immoral cousin considered a severance of kinship ties?

Your female cousin's son is not among the kin whose ties must be maintained, according to the more preponderant opinion. Even if he were considered as such, if a definite harm is feared from maintaining ties with him, then it is not obligatory. Rather, it is permissible to abandon him for more than three days, while considering the benefit in such abandonment. If there is a need to interact with him, he is a non-mahram to you, and interacting with him is permissible within the bounds of necessity. Virtuous and righteous people can be assigned to advise and guide him.