Never actually hunted over there, but lots of ducks are killed in fields(including peanut fields) in several other nearby states, so I don't see why it would be any different here. Don't know about the lakes either, but I would propose you look into a boat, even if it's just a flat-back canoe and a little outboard.
The ducks in the Cruces area are usually gone after Sept/early Oct because that's when the irrigating stops around here.
"A man may not care for golf and still be human, but the man who does not like to see, hunt, photograph or otherwise outwit birds or animals is hardly normal. He is supercivilized, and I for one do not know how to deal with him." Aldo Leopold