Yes, you are correct. The easiest way to think about it is that while the gravity from the earth moves at the speed of light, while it travels toward the sun, it also moves at the same original speed the earth was traveling at.
Te end result is that it appears to travel instantaneously since it ends up at the same place. But this only works for constant velocities - if the earth changes direction suddenly this "correction" will not be correct, since it will still "assume" the earth is still headed in the direction it was heading before.
This is more or less a correct interpretation (the physics are definitely right). Importantly, the behavior is seen in electromagnetism, which is much easier to think about than general relativity, so people should start there.
(But no one will ever read this comment because the topic is 5 days old.)
Te end result is that it appears to travel instantaneously since it ends up at the same place. But this only works for constant velocities - if the earth changes direction suddenly this "correction" will not be correct, since it will still "assume" the earth is still headed in the direction it was heading before.