Can a mother get full custody without a court case or notification to the father.

I am not sure I completetly understand your question. If you are not married to the father, and the father has not been adjudicated as being the biological father of the child, you automatically under the law have full custody and placement rights of the child. The (alleged) dad has no rights to the child whatsoever. If you are married, but not in any legal divorce or separation proceeding, you both have rights to the child. You can however, take the child from one residence to another, or even move away from the dad, as long as you don't "conceal" the child from the other parent. Concealment can be arguably construed as a felony kidnap crime. If you are in the middle of a divorce case  or legal separation, you should file for an initial FCC court hearing to determine temporary custody and placement rights as you go through your case.