Hanging moved further up the list than drug overdoses. Hanging became the second most common method. We don't know where drug overdoses now appeared in the list but it must have been third or lower. We don't know what the most common method was.
You can't replace the "comma + becoming" construction of the original sentence with "which became" or "which has become". The "which" in your versions appears to refer to drug overdoses. The "becoming" in the original construction refers back to "Hangings".