I can't search it up now, but I have seen a derivation that if you work all the way through the math, it turns out your intuition is wrong and in the case of something like a planet orbiting a star, if the star is moving at a reasonably constant speed (ignoring all the relativity details around how to define that for now) it turns out that the planet actually will effectively orbit around the "current" position of the star, even though gravity only travels at the speed of light [1]. Our naive expectation that it would orbit only where it sees the star in the sky right now fails to account for some additional correction terms that show up when you take the dynamics of the situation into account, and it happens to come out in what you may think of as a coincidence to "correcting" the point actually being orbited to what is also the current position of the star. (Technically I think it is still off, but galactic orbits are very, very slow, so the errors introduced by them are also very very small.) You may recall in physics class how the difficulty of understanding situations amped up once you move from statics to dynamics, especially if you took a real calculus-based version of it. In much the same way that even if you use Newtonian gravity, simply knowing the formula may still leave you surprised at quite a lot of what can happen in orbital mechanics.
If the star suddenly disappeared or zoomed off in another direction, it would be a light-speed delay before the planet "noticed" anything, but that generally does not happen, obviously.
I expect the derivation I saw would not be valid for orbits involving speeds close to c, but I would expect the general observation that the effective center of the orbit is in fact not the time-delayed location would still hold.
[1]: Expected for a long time, but fairly recently observationally proved: https://www.youtube.com/watch?v=Pa_hLtPIE1s