I have been repeatedly told they will give you water at any food stand or restaurant, not the carts since they don't have a water fountain or supply of clean ice.
Also, they do allow you to bring in your own water, even in a small cooler with some ice. They will, however confiscate any other drinks you try to bring in (or send you back to your car with it).
Sometimes it depends on who is checking you through security on the size of the softside cooler, but they will ALWAYS check larger coolers to make sure you are not brining in lots of sodas, etc.
I even saw them confiscating a 12-pack of soda out of a cooler recently that a family was brining in with their stroller and small kids. they were told they could keep the water, kids drinks and kids food, but the sodas had to go to their car or be given to the security people.
We always bring a bottle of water in a fanny pack for each of us and they have no issues with it...then just refill them as needed.