Can I leave the state with my child if her father and I aren't married?

If neither party has initiated a filiation or custody action in the county in which you and the child have resided for at least the last six months, there is nothing preventing you from leaving with the child. The problem with doing that is the father can bring an action to determine that he is the natural father of your daughter (that is the filiation part) and seek custody and/or parenting time. Once that case is filed and the papers are served on you wherever you are, you would have to return to Oregon and appear in that custody action or risk losing custody of your daughter to a default judgment in favor of the father. Also, in reaching a decision on custody, the Court could hold it against you that you left the state with your daughter without giving her father any notice of your intent to do so. On another point, if your daughter's father has subjected you to domestic violence or the threat thereof, you can go to the courthouse and seek a domestic violence restraining order (FAPA Order) against him which would be issued within one day without the need to serve the father before the order was issued. To get a FAPA order, you would have to allege that he either assaulted you or did things within the last 180 days that placed you in a reasonable fear of grave bodily harm. If you do go this way, there will be someone at the courthouse who can assist you in filling out the forms and getting them before a judge. One the FAPA order (which would include a temporary custody order) is issued, he would have 30 days to file a request for a hearing so he could contest the facts you assert in your initial motion for the FAPA order. If actual or threatened domestic violence is not a factor in your relationship, your best course is to go see a lawyer who does domestic relations cases.