Gravitational waves and gravitational influence are different beasts.

Gravitational influence is otherwise known as the curvature of space-time, and can be described as an effect of geometry - the mass of the Sun curves the space-time around it, and the Earth follows that curvature. This curvature does not need to "propagate" as such. The Earth orbits the place the Sun is "now" (approximately), not where it was 8 minutes ago.

Mass tells space how to curve, and space tells mass how to move.

General Relativity tells us that in the case of weak fields (like our Solar System), where any changes in accelerations between bodies are very small, the curvature of space-time "telegraphs" the instantaneous positions of those bodies to each other (so the Earth already knows where the Sun was 8 minutes ago, and the curvature points the Earth towards where the Sun will be "now", if the Sun hasn't suddenly accelerated in the meantime). So, in the absence of any sudden changes in acceleration (known as "jerk"), the curvature of space-time neatly extrapolates (approximately) the instantaneous position of the bodies in question to each other.

But, if there is any change in the acceleration of a body ("jerk"), that extrapolation then "misses", and the result is the propagation of gravitational waves across the curvature, at c.

It might sound strange that gravitational influence works as if it is instantaneous, but if the information is already contained in the system as to what all the bodies involved are doing, and none of them changes what they are doing, it kind of makes sense. Only when something changes what it is doing are gravitational waves produced, as a result of the "miss" in the extrapolation.

Sorry if I haven't explained this very well.