Is the one in Carlsbad Cavern... which is not filled with water, I don't think.
The one I saw, as a kid, I'm not sure of... such a distant memory. I can picture it in my head and have searched the net but haven't seen it. Like I said.. .I'm not even sure it was in NM.
I read a Clive Cussler novel some years ago that took place on an underground river that emptied into the Gulf of Mexico. I remember when reading it how 'fantastical' it all was only to discover after reading the novel that although embellished for the novel that underground river does exist.