Your gifs have that banding because they can only display 256 colours. Limitation of the format unfortunately. Don't worry, it doesn't affect the experience of the gif much :)
Well you can use dithering to give the illusion of a larger range, not sure if whatever you used to capture it would have the option to do so, or how good it would look if you do. Unknowns! Best of luck trying to fix it but don't spend too long, they're fine as-is for showing development :)
