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As we know Earth rotates around Sun and Sun around Milky Way but then Sun must have some velocity/angular momentum (I don't know that much physics terms).
So as Earth revolves around Sun, it must adjust its orbit as Sun also revolves.
Is it so? Or am I missing something?
Earth is bound gravitationally to the Sun. So it is orbiting together with the Sun around the center of the Milky Way.
This takes one galactic year.
Hence your presumption is correct.
Seen from a non-rotating system relative to the Milky Way barycenter, Earth's orbit isn't an ellipse, but ressembles a distorted helical trajectory. Distorted, since the orbit around the Sun isn't perpendicular to the orbit around the center of the Milky Way.
It is a composite movement.
Earth does not need to adjust anything: Earth's reference frame for orbiting is Sun, almost alone. You can take into account Jupiter, Saturn, Moon and that's pretty all. If you want to do things really really precise you use General Relativity and take into account Venus, Mars, Mercury, Uranus and Neptune. If you want even further precision you put into Dwarf planets and asteroids.
Only once you have all of that into account, you need to think "hey, there are other gravity forces out there", because Solar System's movement is almost rectilinear, 'cos the enormous radius of the orbit. So it affects almost exactly the same to Sun and Earth.
In resume, you can simply think "Earth orbits Sun, and idependently of that, the System moves towards the Apex".
About angular momentum, yes, there is a good amount of angular momentum in those movements. And also velocity: the sun moves through Milky Way at 72,000 kilometers per hour.
