I am a young mom and would like to give up custody to my baby to another family member. How do i go about that?

You need an attorney to help with this so that you fully understand your rights. this is not a simple thing.

You don't say anything about the father's role here. Whether you like it or not, if the father has not had parental rights terminated, he has a say in this decisionmaking. I'll presume for now that both you and the father both want to give up custody.

There are two legal events here. The first is called a voluntary termination of parental rights. This legally removes all of your rights and duties to the child. After this is done you ordinarily won't have any right to make decisions for the child, visitation rights, or a duty to provide support for the child. When your rights are terminated, the child will go into foster care. You and your cousin can ask the court that she be designated as the foster parent, but she will have to work with the child services agency in your area to be ok'ed as a foster home. The court has the final say, based on the child's best interests, where the child will go.

The next step is adoption, whereby someone else becomes the legal permanent guardian of the child. You can recommend to the court who will get custody of the child, but understand that this is only a recommendation. The court will decide, based in the child's best interests, what should happen to the child. In short, since you will have at that point surrendered your parental rights, you don't get to decide what happens to the child.

This is very serious stuff and you should consider your decisions carefully. Please consult with an attorney and find out if TPR and adoption is the best course that serves the best interests of the child. If the father has not had parental rights terminated, this is going to get sticky because if (a) he doesn't want the baby going to your cousin and (b) you terminate your parental rights, then decision making for the baby rests 100% with the dad.

If you and the father are in agreement that this is what you both want, get an attorney to help you through this.