You might enjoy Kurzgesagt's video on moving the Sun (and with it the whole solar system) they propose a method that would theoretically let us colonize the galaxy in a reasonable (but not trivial) time frame.
To your exact point though, since we're moving the whole solar system, everyone would be living their whole lives on the/a ship.
There are rumors that we should pass close to some other stars over the next couple million years. If we live that long that may be how this goes. Push the limits of stellar travel to visit the new neighbor, check out how habitable it is, start a diaspora. I don’t know how fast things like that will happen. Maybe you have months to decide which side you want to be on. Maybe you get months. Or a generation. But eventually they will get too far apart to visit again, except to share stories between us.
To your exact point though, since we're moving the whole solar system, everyone would be living their whole lives on the/a ship.
https://www.youtube.com/watch?v=v3y8AIEX_dU