(a) No person elected to the office of Mayor or to the City Council shall, during the term for which he was elected, be appointed to any office or position of emolument in the service of the city.
(b) The Council or any member of the City Council shall forfeit his place on the Council, as required by Article 11, Section 11 of the Texas Constitution, if he becomes a candidate for nomination or election to any public office other than a place on the City Council or if he becomes a candidate for election to any different place on the City Council which requires him to take office prior to the end of his elective term.
(c) The Mayor or any member of the City Council affected by subsection (b) shall carryover or holdover in his service in the position as permitted by Article 16, Section 17 of the Texas Constitution.