Try the following solution - before saving your animation, in the Animation Properties tab, enable Do not merge palettes.
Each GIF picture has an internal color palette attached to it. This color palette can contain up to 256 colors. In a GIF animation, you can have single palette for the whole animation or separate palettes for each frame. Each palette increases the size of animation file.
By default, color palettes of individual frames are merged into one palette. This method allows to greatly reduce the size of animation file and it also provides good quality in most cases. Sometimes, however, when each frame in your animation contains very different color, it is better not to merge individual frame palettes into one.
Please note that even after implementing the above suggestion, your GIF images might still look a little bit grainy. This is due to limitations of the GIF image format which can hold only 256 colors per image frame. This has nothing to do with the capabilities of Easy GIF Animator.