Based on what you have written, just signing over your rights may not dismiss your responsibility to adequately provide for your children. There are many factors which a Court will consider when determining what is in the childrens' best interests. You should consider consulting with a local law firm to gain a better understanding of the process and aspects with relinquishing parental rights.
If my local law firm can assist you with this legal issue or any other legal matter, please contact us at firstname.lastname@example.org to arrange for an initial consultation. The above response given is not intended to create, nor does it create an on-going duty to respond to questions. Additionally, the response does not form an attorney-client relationship, nor is it intended to be anything other than an educated opinion. It should not be relied upon as legal advice. The response given is based upon the limited facts provided by the person asking the question. To the extent additional or different facts exist, the response might possibly change.