No, the methane isn't being turned into chlorinated organic compounds. The reaction mechanism is CH4 + Cl --> CH3 + HCl. (Caveat: I don't think the side reaction CH4 + Cl --> CH3Cl + H occurs, but I need to confirm.) The chlorine ends up as hydrochloric acid, which washes out in rain. If you object to the acidity, note that the electrolysis of seawater was leaving behind the sodium ions as sodium hydroxide, so there would be no net increase in acidity. The methyl radical is also produced by the natural oxidation of methane by OH radicals, and would continue along that natural oxidation chain.

The link I referenced (and the paper it cites) do not, as far as I can tell, discuss any chlorinated organic products.