They are required by California state law to provide free cups of water upon request, this has been confirmed many times by guest relations, they are also required to give you the smallest size above a child size, if you request it. If they try to refuse, simply point this fact out and that guest relations has confirmed it, they change their mind real fast. Many will give you the new larger cups as well, just always ask for a large cup of water. Please note that under California law the soda machines water must be filtered, please note that regular water is usually under iced tea or lemonade, the one under sprite would be carbonated, but now there are so few self service machines that you shouldn't need to worry about it too much.
I especially like the Rancho de Zocalo spot because they have water outside in the dining area. And it is more central to the overall park. We usually pop inside though and fill our bottles with ice and then get the cold water in the outside dining area.
On top of the above mentioned you really can't walk very far in either park without running into a water fountain, which from my experience at least has always had cold water
Last time I was there the water from the drinking fountains had a funky flavor. I could be picky but it tasted weird to me.
