Does the Earth wobble as it spins?

Indeed it does. Astronomers have been monitoring its wobble over the last century and have found that the axis of the Earth dithers in a roughly circular path called the Chandler Wobble. At the North Pole of the Earth, if you were standing there and plotting where the axis emerges on the surface of the Earth, its track would execute a circular loop about 20 feet in diameter. This is reflected in a 0.1 arcsecond circular wiggle in the paths of stars during the course of a year. This wobble is not the same year after year and can vary from 10 feet to 50 feet over the course of several decades. The largest amplitude was recorded, I believe, in 1910. The cause of this has many sources including rearrangements of mass deep in the Earth's interior.