The writer is being expressive by using more than one word. In part, she is using "shiny and shimmery" because of the alliteration--the words begin with the same sound--which sounds nice. "Shiny" emphases reflection. "Shimmery" emphasizes motion and change. A mirror is shiny, but not shimmery. Shiny cloth on a dress moves when the person wearing it walks, so it is shimmery as well as shiny. If you can see your face reflected in something, it is shiny, but not shimmery. If you just see bright flashes in motion, it is both shiny and shimmery. "Sway" and "wobble" both mean leaning back and forth. "Swaying" can be graceful, smooth, slow, and intentional. Dancers "sway" to the rhythm of the music. "Wobbling" is faster. It often suggests clumsy, imperfect balance. (I can stand on one foot, but I wobble when I am doing it.) However, it doesn't mean that here. It just means a faster motion than when she "sways."

Shiny is more like when you see light shine on a metal, where shimmery is like when the light reflects back and forth on a surface like on a diamond or when the sun shines on the water. Wobbly is like when someone is about to fall over and lose their balance, where swaying is moving slightly back and forth, but you are in more control, and are not as likely to fall over.