Well, real orbits are not perfectly circular (and many stable orbits are far from circular) so the distance (temporal and spatial) certainly changes.

That's true - you'd have a possible gravitational blue/red shift at different points.

Gravitational waves aren't how gravity propagates, however. They are really only relevant in exotic situations like mergers of massive compact objects, where spacetime is warped enough that orbits aren't stable and the bodies spiral inwards, radiating their excess gravitational potential energy in the form of gravitational waves.

If gravitational waves are not how gravity propagates, how does gravity propagate?

In general relativity gravity is curvature of spacetime caused by mass-energy. Changes in curvature propagate at c, the speed of causality. Gravitational waves are a special case of those changes predicted by GR, inspiraling massive objects making the spacetime around them ”ring” like a bell.

An eventual theory of quantum gravity may entail another viewpoint of gravity as a quantum field and mediated by a hypothetical graviton. But as of now we don’t have a good idea of what such a theory might look like; unifying QM and GR is famously one of the great unsolved problems in physics.

"In general relativity gravity is curvature of spacetime caused by mass-energy. Changes in curvature propagate at c, the speed of causality."

I really hope you meant speed of light there.

Regardless, this thing is still a mystery to me. Why, if gravity propagates at c (or, better perhaps: Changes in the space-time curvature, which is caused by mass-energy, propagate at c), why doesn't the earth spiral into the sun or away from it? If the earth is attracted to the point where the sun was eight minutes ago and not to where it is right now, it's orbit should not be stable. It's a very big difference compared to the "gravity is instant" way of Newton.

The speed of light in a vacuum is c, which is the speed of information, or the speed of causality

I used the term "speed of causality" on purpose; that's what c is really about. The fact that the speed of light in vacuum happens to be c is just a side effect of photons being massless.

Thank you again for your of answering my questions. So.. well.. What is this speed of causality then? Can we put a number on it?

Besides this all, I still can't get anyone to give me a clear cut and dried answer on my original question. If gravity is instant in the Newtonian model, and that gives us stable orbits and all that, how does that translate to the Einsteinian model, where there's an upper speed limit of c? So, why would that even work, if the gravity of the sun takes 1au/c to get here and be felt by the earth? Or, is gravity somehow "above" this speed limit?