giving up fathers rigth

The court is understandably more concerned with the child's rights than with yours. The child has a right to be parented and a right to be financially supported to the degree that the parents are able to do so.

This means that the court is strongly in favor of you spending some amount of time with the kids. There can be a number of reasons why this is not a good idea in your case, and if one of the reasons apply, and neither you nor the mother is in favor of you spending time with the kids, the state will generally not check on you to see if you are spending your residential time with the kids.

Child support is another matter. Child support is just what the name implies, money that is to go toward meeting your children's needs for food, clothing, medical care and so forth. Many fathers think it is a fee they pay to see their kids, and if they stop seeing their kids, there is no further need to pay. This is, of course, not true. The child continues to need to eat and be clothed whether or not Dad spends time with the child.

So in other words, if you want step out of your children's lives, and not spend time with them any more, and the mother agrees, the state will not approve, but is not likely to interfere. Child support is another matter.

The only way you can legally relinquish your parental rights and obligations is if someone else is willing to step in and assume them from you. In practical terms, this means that if the mother acquires a step-parent for the kids, and that step parent is willing to adopt the twins and become responsible for them as an adoptive parent, you can then agree to release them for adoption by the new adoptive parent. There is,of course, a legal process to go through for this.