That sounds awful-I haven’t seen that livestock problem here-livestock must be confined to your property with a fence-if your cows/goats/etc get seen out, you get fined bigtime.
I live on a cliff over the water-which is patrolled and tested every day by fish and game for not only livestock crap, but pesticides and weedkillers that are used by people who ignore their deed restrictions. Those fines run as high as 10K, for obvious reasons, but the iodine tablets are a good thing to keep anyway.
New Mexico is a “fence out” state which means you must fence cattle, sheep, horses, etc. out. From the NM Livestock Board:
http://www.nmlbonline.com/documents/fencing1210.pdf
Also, national forests, including wilderness areas, allow cattle grazing. There are very few streams that are not polluted, at least somewhat, by livestock in the forest.