Yes, I think these the county clerk and chancery court would be considered state/local government and the rule would apply if the conditions below are met.
(iv) A check drawn by a state or a unit of general local government and deposited–
(A) In an account held by a payee of the check;
(B) In a depositary bank located in the state that issued the check, or the same state as the unit of general local government that issued the check;
(C) In person to an employee of the depositary bank; and
(D) With a special deposit slip or deposit envelope, if such slip or envelope is required by the depositary bank under paragraph (c)(3) of this section.