I think it's actually more complicated than that. The strength of the sun's gravity goes as 1/r^2. Move it to the side a bit, and the direction of the gravitational field changes slightly. This direction change goes as 1/r, by the small angle approximation. So from your description, we'd expect the size of the ripple to go as 1/r^3. But in the case of actual gravitational waves, the field strength of the ripple goes as 1/r. That's why we can detect gravitational waves from distant black holes, but we can't detect their static gravitational fields, which go as 1/r^2.

Electromagnetic waves are similar. The Lienard-Wiechert field formulas [1] have 2 terms. The first term describes the delayed field, and is proportional to 1/r^2, while the second describes waves and is proportional to 1/r.

[1] https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_...

Unfortunately that’s not how they behave at all. See some of the references posted by others in other comments. Intuition doesn’t really work at all here.

Agree with jleahy.

I would add that considering counterfactual situations like "if the sun were to move instantly to the right" is of limited utility. That cannot happen (violates conservation of momentum). (Granted, other thought experiments have been quite influential in this field...)

Apart from some tidal forces we probably wouldn't feel a thing but rather just continue following the free falling trajectory.

Gravitational waves are also different from changes to the amount of gravitational force. This can be seen from the fact that gravitational waves affect the distances between objects, whereas the sun which exerts far more gravitational force has no measurable effect (at least, to my knowledge rotating objects don't change length noticeably).