Q: I live in Oregon and wondering if I can leave the state with my newborn.

A: There is no legal reason you can't pack up and leave with your baby. The problem however is not whether you can legally leave but rather what will happen once you do leave. For a period of time after you leave the State of Oregon will still have jurisdiction over your baby so either man could file in the State of Oregon asking for things like the establishment of their parental rights, parenting time, or custody. A court could order the that baby be turned over to the "father" if the court awards custody, but that could happen if you stay in Oregon so the risk isn't any greater. As a practical matter it is extremely unlikely that a court will order a newborn to be placed with the father and taken from the mother. The only time that happens is if the mother simply can't take care of the child or is having problems like mental illness that isn't being properly managed or is involved with criminal activities or substance abuse. Absent these issues you are more likely to have to deal with the court ordering some form of parenting time.

Historically the husband of a married women was always deemed to be the legal father. Then Oregon Law came up with the exception that this might not be the case if the husband and wife were living apart at the time of conception. It is hard to say if your relationship with your husband will conclusively establish him as the father even if he isn't the biological father under current law. I think you should not assume that you can rely on this assumption and you should consult with a lawyer, discuss the facts of the situation, and let the lawyer research the current Oregon Law on this issue.

Also I have a concern that if someone is really determined to challenge the paternity determination based on the finding that a women was married and residing with her husband at the time the child was conceived, so the husband is presumed to be the father without considering an actual DNA test, the other person, the supposed biological father, could argue that this is unconstitutional and maybe win. There have already been cases that went to the US Supreme court, dealing with other issues, that held that a biological parent has a constitutional right to raise their child. This is the case I am thinking of: https://www.law.cornell.edu/supct/html/99-138.ZO.html