First, don't worry about it.
The axis of the Earth does change naturally, but imperceptably, all the while. It is already a long way off "square" anyway. It is that tilt that gives us summers and winters in fact.
Even an event such as the movement of the Earth's tectonic plates is so tiny on the global scale of things that it makes no difference. The Earth's crust is always on the move. Constantly. When I fly from London to the USA, the journey is about 2-1/2" further each year. That's because the American continental plate is still moving away to the West.
Think of the Earth's surface as being a very thin crust on a rice pudding, but with several separate areas of crust. They float around separately because of the convection currents of the hot rice pudding underneath. Where the crust of the rice pudding is thinnest is where the liquid underneath is rising, and then the "plates" of the crust spread out. The problem happens where two lumps of crust bump into each other or slide against each other. That happens all the time. The San Andreas fault in California is one of those places for example. That has moved a long way in the last thousand years alone. But it hasn't made any difference to the axis of the Earth. The movement is too small to make any difference. It would take a huge thump from an external force to do that! The whole of the Rocky Mountain range was created by the Pacific plate burrowing beneath the continental plate and pushing, and crumpling up its edge. That didn't change the Earth's axis. Now we are talking BIG movements, right?
Press the palms of your hands together as hard as you can. Then slide one hand against the other. It will suddenly jump a little way. That's the same as what these plates are doing when they move suddenly. Let them move little and often and there is no big deal, let the pressure build up and then suddently break, like your hands just did, and there is a shock. An earthquake, instead of a little tremor. But whether it is sudden, or in little movements, the total distance moved is always going to be the same.
And, those "little" movements (on the global scale of things) are not going to move the axis of the Earth. Well, perhaps a little over many, many millions of years.
If anyone wants to get serious about these things, then move out of California. There's much more danger there than little movements changing the axis of the Earth.
Rod