The angle of polaris and the earth axis are same has about 1/4 degrees difference.
But our glober friends claim that the earth bowing a side 5 degrees for summer and bowing to the other side for winter. It makes a total of 10 degrees. however, the angle of polaris don't change to the earth more than 1 degree difference. This means the earth looking the direction different summer and winter that has 10 degrees difference.
if the polaris is moving at a speed where the earth's new direction or we need to throw away the system.
The axis doesn't physically change, it does relative to the sun however.

Polaris is a very far distance away(433.8 lightyears, or 2.55 * 1015 miles.) So going in orbit around the sun(a maximum horizontal change of 185,900,000 miles) will only give a very slight change in the angle of Polaris.
The earth axis is phsically impossible.
Because which pole is near to sun, it must continue to be near; which pole is far to sun, it must continue to be far. You are drawing the opposite it but it opposite of the saving the momentum.
Look to this shape:

Watch a spinnig of a whirligig. Which way is it turning around, it is usually the same side is bent to the center of the way. You are claiming opposite. So you need a very strong argumant to say except imaginary evidences.
Well, the reason that doesn't happen is because the earth is
massive. Conservation of angular momentum and rotational kinetic energy would state that it would take a humongous amount of energy to turn the earth's axis while it is spinning. That's why axial precession happens.(much like holding a spinning bicycle tire) Your whirligig comparison isn't akin because that doesn't have extremely high amounts of mass. However, you are sort of right. The earth's axis does change
very slightly. Over a period of 13,000 years, it will turn 47 degrees counter-clockwise due to tidal effects of other celestial bodies.
Imagine a spinning ball in space. It will spin forever unless a force is applied to it, right? It won't randomly start moving in a direction, right?(Conservation of translational energy) In the same idea, it won't start rotating randomly(Precessional force and conservation of angular momentum)
Now, if you don't know what precessional force is, take this experiment. If you have a bike, take off the front wheel and spin it very quickly. Hold the axes with both hands and try to rotate it. The force will cause your body to turn in a direction perpendicular to torsive force you are applying. Imagine doing it with a larger, more massive, and faster spinning object.
Hope this helps.