Whenever a person, committed to the care of the State Department of State Hospitals or the State Department of Developmental Services under one of the commitment laws which provides for reimbursement for care and treatment to the state by the county of commitment of the person, is transferred under Section 7300 to an institution under the jurisdiction of the department where the state rather than the county is liable for the support and care of patients, the county of commitment may have the original commitment vacated and a new commitment issued, designating the institution to which the person has been transferred, in order to absolve the county from liability under the original commitment.