New York is the nation's third leading producer of dairy products. I don't see any dairy farms around here sending it to landfills, either. Most of is stored and then spread on the corn and soy fields the dairy farms use to feed the cattle. If there's too much, then some of the farmers will sell it off or give it away to gardeners.
Either way, some of the thousands of acre farms will build a storage facility (usually a concrete above ground pit with three side) covered in tarps.