Embodiments provide a method and apparatus for upgrading a hydrocarbon feedstock. According to at least one embodiment, the method includes (a) supplying a hydrocarbon feedstock to an oxidation reactor, where the hydrocarbon feedstock is oxidized in the presence of a catalyst under conditions sufficient to selectively oxidize sulfur compounds present in the hydrocarbon feedstock; (b) separating the hydrocarbons and the oxidized sulfur compounds by solvent extraction; (c) collecting a residue stream that includes the oxidized sulfur compounds; (d) supplying the residue stream to a coker to produce coker gases and solid coke; and (e) supplying spent adsorbent including residual oils from the adsorption column to the coker for disposing the spent adsorbent after completion of an adsorption cycle.