I saw some take and toss cups at Target just this week that had screw-on lids. They also said they were spill proof. I think they were a newer version of the ones with the snap on lids. I didn't get them though, so I can't say how well they work.
My kids were a bit older than yours and we took lots of breaks, so we didn't bother with sippy cups in the parks. We just used hydration backpacks and drank water in the parks. (You also would get a drink if you are buying kids meals.) They had regular sippy cups in the room and drank juice and milk when we were there (morning, afternoon nap, and evening). That way we didn't have to worry about carrying/losing the cups in the parks and having them get nasty in the heat.