Hi Emma, sorry to jump in. Having looked at your animation I thought it looked pretty good and I thought initially it was an intentional style choice. I think what you are referring to as 'pixelated' is actually 'colour banding' and is quite common with the GIF format. The GIF being an older format has inherent limitations. For example it can only support 256 colours and uses various (compression) methods to compose the image you see on the screen. I have never used 'Screen to Gif' so I am unsure what compression methods they use for their GIF output. But in Photoshop for example you can do things such as reduce amount of colours or apply 'Dither' (a way to make colour banding look smoother). My guess with your image is that the 'screen to Gif' software does not apply Dither which is causing the colour banding. I could be wrong but that's my best guess. I would suggest using a different tool to create the gif. Using storyline to create the animated GIF is probably not the best idea, but if you are limited by software a work around might be to record short screencast of the animation cropping out any background (maybe you can use articulate Peek?) then you are free to use other GIF generator tools on the web (https://giphy.com/create/gifmaker).
I apologise if you know all this stuff already!
Andy
p.s. I'm English, hense the spelling of colour and apologise ;-)