See how tricky this is? You have just claimed local motion (relation to the Sun) but used a stellar reference frame (Draco) as your proof. Remember the stars are “outside” the SS. So to prove the earth wobbles relative to something “inside” the solar system you would have to show it wobbled relative to the sun or moon or planets. Currently no astronomer does this, and for good reason, everything in the solar system has a much higher rate of proper motion than stars outside the solar system. But there are some ways the truth can be derived, such as the lunar equations, but to date the paradigm is so comfortable, no one is even asking the question if precession might have an alternative cause.
The official method of measurement, Very Long Baseline Interferometry (VLBI), measures changes in earth orientation (the big one being precession) relative to quasars, energetic stars far outside the moving frame of the solar system. But the oversight is the luni-solar theory does not consider reference frame changes, meaning the dynamicists assume all changes in earth orientation are due to local dynamics (like the gravity of the sun and earth tugging on the oblate earth, earthquakes, tidal motions, etc. etc.) and therefore they effectively use a static solar system model. It has been this way for hundred of years - thanks to Copernicus who said the sun does not move.
The observable of precession (stars moving across sky) is obviously real and ongoing – and clearly most people believe this to be the result of a wobbling earth – but we can find no evidence the earth wobbles more than a few arc seconds relative to objects within the solar system.
Now obviously to claim the earth does not wobble at all is heresy. I do recognize nutation (roughly 6"-9" of bob over an 18 year period, exactly in synch with the geo-location of the moon) and Chandler wobble (about 10 feet of axial reorientation), but none of these approach the magnitude of precession, which exceeds 50"p/y – that’s big!
Needless to say I have taken a lot of flack for suggesting that the precession observable has a largely non-local cause. But unless some very compelling evidence arises to the contrary I am convinced the solar system moves. Eppur si muove.
Chinese Philosopher – Lao Tzu