Yes, Hi!
I want to know how to go about relinguishing my rights of my unruly, delinqent child and not lose custody of my well behaved child?

Go to parenting classes and learn how to be a good parent.

Edited to add: Unless someone is going to adopt your "bad" child, then you cannot.

Sorry we can't pick our kids,

What exactly do you mean by " delinqent " ?
Is this child in constant trouble with the law ? How old is this child ?