We generally travel with our non-disposable water bottles on a carabiner (can clip it to almost anything!). At all WDW Parks, any "fountain beverage service counter" has free ice water. Asking politely, I have found most will give you a large cup or 2 of ice water.
We will then split it between the 3-4 of us and there isn't the weight of a full bottle for anyone to carry, especially after drinking all or most of it.
During busier times, they have small cups filled on the side that they will then quickly pass to you, so try to avoid during busier times unless you're going to stand in line for something else.