Congratulation on the birth of your child.  I can understand that this may be a very stressful time.  Oregon is a state that does indeed have a set of laws recognizing the right of a child at least 16 years old to file for emancipation.  Generally you need "a plan" for supporting you and your child that does not include public assistance.  A place to stay, a job a method of support, a plan for attending school, etc.  Here is the law:

§ 419B.558¹

(1) The juvenile court in its discretion may enter a judgment of emancipation where the minor is at least 16 years of age and the court finds that the best interests of the minor will be served by emancipation. In making its determination, the court shall take into consideration the following factors:

(a) Whether the parent of the minor consents to the proposed emancipation;

(b) Whether the minor has been living away from the family home and is substantially able to be self-maintained and self-supported without parental guidance and supervision; and

(c) Whether the minor can demonstrate to the satisfaction of the court that the minor is sufficiently mature and knowledgeable to manage the minors affairs without parental assistance.

(2) Upon entry of a judgment of emancipation by the court, the applicant shall be given a copy of the judgment. The judgment shall instruct that the applicant obtain an Oregon drivers license or an Oregon identification card through the Department of Transportation and that the Department of Transportation make a notation of the minors emancipated status on the license or identification card.

(3) An emancipated minor shall be subject to the jurisdiction of the adult courts for all criminal offenses. [1993 c.546 §136; 2003 c.576 §452]

Also look here:

http://courts.oregon.gov/Marion/docs/MaterialsAndResources/AppForJudgmentofEmancipation.pdf

Good luck.