Dog Logan is also Elizabeth's son.
He has claws, like John and James Howlett, but John as far as we know had no healing factor. It makes sense if Logan had a proto-healing factor of sorts, never being hung over despite being a raging alcoholic and so on, but assuming John was indeed Howlett's biological son, the claws are definitely from Elizabeth.John Howlett Jr. is still alive.
He refused to die, so his parents drowned him in a lake or swamp. Poor kid's been drowning since 1895.