One cosmic year is at least 225 million years long, that's the time it takes for the sun to rotate around the galaxy. Also, consider that the other stars that we can see - all in our neighbourhood of the galaxy - are also moving in almost the same speed.
The stars do move indeed, but it takes hundred of thousands of years before the movements gets visible with the naked eye.
At the other hand, there is also the precession causing the stars to move in circles relative to the spin axis of the earth. The current polar star hasn't always been the ideal pole star, ref wikipedia