"trying to spin really fast"
Do you need to change gears at that point? If you are in too easy a gear you may be pedalling at a faster cadence than your body can go and still maintain a steady posture on the bike.
Like everything Lisa mentioned, this can change too as your muscles adapt and your body learns new movement patterns.
For now you might feel more steady if you get into a slightly harder gear. But be aware that as things change your gear choices will also change. It's all very cool! (my comfortable cadence is much higher now than it used to be)
"If Americans want to live the American Dream, they should go to Denmark." - Richard Wilkinson