Can she still take my son without my permission to another state even though we are not married?

Marriage is not the issue. You are the biological father and have rights to see the child as well as responsibilities to take care of it. She should give you notice, typically 30 to 45 days before leaving, and then you need to go to court to assert your rights.

This needs to be done as soon as possible. You have a clock ticking down now. It will be difficult after she moves away. It needs to be addressed immediately.

If there is not a court ordered custody agreement in place, she certainly can take the child without your permission. I suggest seeking the aid of an attorney for an emergency order to be put in place.