First: Understand that, unless specific arrangements have been made otherwise, the courts won't be checking to see if you moved or if the father cares. Ultimately, 98% of divorce actions are filed by one or the other parent, NOT the state.
To answer your question, yes you can... Unless:
The divorce decree specifies that they can't.
Or
The father gets wind and file with the courts as a hardship/violation of visitation rights.
Understand something else: Should you move to another state and then want/need to take legal action against the father, you'll have to go back to the original state to really even stand a chance, and that state may not rule that the father has to pay your expenses to do so.
Consider how difficult it could be to either pay a lawyer to file and appear for you, or to go there, file (and pay filing charges), arrange to have the father served, then, when the court date arrives, go again, be in court (staying at hotels?), and then arrange for enforcement of the ruling.
To answer your question, yes you can... Unless:
The divorce decree specifies that they can't.
Or
The father gets wind and file with the courts as a hardship/violation of visitation rights.
Understand something else: Should you move to another state and then want/need to take legal action against the father, you'll have to go back to the original state to really even stand a chance, and that state may not rule that the father has to pay your expenses to do so.
Consider how difficult it could be to either pay a lawyer to file and appear for you, or to go there, file (and pay filing charges), arrange to have the father served, then, when the court date arrives, go again, be in court (staying at hotels?), and then arrange for enforcement of the ruling.