Admittedly, I have only ever heard this version of the story in books. My grandmother when I was little told me that La Llorona was "a crazy woman, who beat her baby to death against a wall." My grandfather described her as an old woman with long terrifying fingernails and as much of a "witch-hag" description as you can envision. My father always describes her as "an old woman in a black dress." There was never any sense of her being restricted to waterways; part of what freaked you out as a kid was the understanding that she might just spontaneously show up inside the house at night.
Moreover, her paranormal activity torments adults just as readily -- and very typically those who have been misbehaving in some way. My great-grandfather had such a ghost sighting when he was driving home drunk one night (she appeared in front of his car; he slammed on the breaks but should have hit her nonetheless; yet, she vanished!) I'm not sure what I was doing when I saw her. An excellent collection exists in the small-press book The Weeping Woman: Encounters with La Llorona compiled by Edward Garcia Kraul and Judith Beatty. She's often not weeping at all; it therefore puzzles, just how she's identified. But in these stories, without fail, she is not a mourning ghost looking for children -- in fact, she's practically Krampus, the German demon who punishes naughty children (and sometimes adults) by abducting them or flogging them.
Despite her function to amuse adults by scaring their children with tales of her mindless malice, she's actually somewhat on the heroic side. Kind of like how some people ascribe pure malice to the Intranquil Spirit, with whom La Llorona could easily be matched -- but while her ways are (we might say) a little anti-social, ultimately her actions are aimed to be beneficial. This web page has a few tales, in which she does things like keeps children from running away from home, and the Library of Congress seems to make efforts to keep a thorough collection of the folklore.