The father of any child under the age of twenty-one years and not married, if the mother is dead, or the mother of any such child, the father being dead, whether the father or mother is under the age of twenty-one years, or of full age, may by deed executed and recorded according to law or by last will and testament, made and probated according to law, dispose of the custody and tuition of the child while he remains under the age of twenty-one years to any other person, in possession or remainder. No deed is valid unless signed by both father and mother, if both are living and no such deed, except a deed to an agency or department of this State authorized by law to receive or place the custody of children, is effective unless approved upon petition by a family court or family court judge of this State. Nothing in this section may be construed to abrogate, lessen, or interfere with the right and duty of a court of competent jurisdiction at any time to transfer and assign the custody of a child for its best interest.