For places that have drink stations, you can walk up and fill your water bottle with water. There aren't that many places like this in the parks. Most places fill the drinks behind the counters.
The only one I can think of off the top of my head is Restaurantosaurus at the Animal Kingdom, because I've done it myself. We did it plenty at the resort.
I used Crystal Light/koolaid packets on one of my last trips, and while it covered the taste fine, I got sick of it after a while.
I also found that the ice helped mitigate the bad taste from the water fountains, so when I did order a fountain drink, I would pour the ice into my water bottle after I finished.
This year, I'll probably bring some flavoring packets, use lots of ice, and I bought a couple of filtering water bottles to take. We'll see how it goes. I don't like the taste of most bottled water either, so it's really a no win situation for me.