I'm reading Stephen Webb's book (If the Universe Is Teeming with Aliens ... WHERE IS EVERYBODY), and he describes how a partial Dyson sphere can turn a star into a spaceship which blew my mind (just cover all but one side, the released energy will push it the other direction). Imagine doing that at the Galactic scale.

>just cover all but one side, the released energy will push it the other direction

What is to stop the star from just crashing into the sphere?

In the book he discusses the shkadov thruster which reflects the energy so the shell would move, too.

Obviously if it's not reflective you could absorb the energy and use it as needed

https://en.m.wikipedia.org/wiki/Stellar_engine