If the father does not sign the papers, then your legal rights as a parent should not be terminated and you should be allowed to parent the child. The adoptive parents could fight to terminate the father's parental rights, and if the father of the child doesn't resist then his parental rights will be terminated within a month or two after the child is born. But if the birth father fights termination of his parental rights the adoptive parents will have to prove in court by clear and convincing evidence that he is an unfit parent. This is difficult and expensive to do, and the adoptive parents may not be willing to take on the financial and emotional risk. So if the adoptive parents aren't willing to fight the father, you might end up parenting even though you may not want to do so.
If you are not fit to parent, then it is likely that you son will be placed in foster care, but only if the child's father is also unfit to parent. The state system will likely look for a relative to place the child with as foster parents. The state is required to provide all services that are reasonably likely to help you and the child's father cure any parenting deficiencies that you have.
You yourself need legal representation and the adoptive parents should be willing to pay your legal bills as part of the adoption.
DISCLAIMER: This answer does not constitute legal advice nor does it form an attorney-client relationship between the questioner and responder. You should seek competent legal counsel in your jurisdiction to address any specific legal concerns you have.