I've been watching a lot of amateur video from the Disney parks recently and most of it is appalling. Nobody knows how to hold the camera still! I admit, with small modern cameras, that it is pretty hard to do. My ancient Super-VHS camera was so big I had to carry it on my shoulder (you remember those things). They were not overly portable but they helped you hold the camera still!
Anywho, the Steadycam is a cheap device that will help you keepy your camera moves clean and smooth.
Unfortunately, Disney might not let you bring it into the park, because of the giant counter-weight.
But here's an idea. Rent a stroller and hook your camera to it. Or buy a mono-pod. They are cheap and super-easy to use and collapse into a very portable size. Even if you just leave the mono-pod attached to your camera while you move your videos will be smoother.
Or ... buy a tripod! You can leave the tripod connected to the camera but collapsed. This will also give you some extra weight that will help you smooth out your camera moves.
Here's the simplest idea. Quick walking with the camera on! You can't do it with this little cameras. What you do instead is stop moving, shoot a few seconds of video, walk further, stop moving and shoot some more, stop shooting, walk to a new place, shoot some more, and so on. Your video will look 1000% better with nice happy edits instead of jerky walking motion.
There is a guy named TJ at the Window To The Magic podcast who does a great job on his "Disneyland Update" video podcasts. Watch those to learn how to do it.
Please, Disney videographers, learn to hold the camera still!