We see the Sun about eight minutes after light left it. Presumably
this means we are also experiencing the Sun's gravity 8 minutes after
it 'left' the Sun.
No, not in the way you mean anyway.
I'm sure you've seen those diagrams where they illustrate gravity by placing a large mass like a shotput in the middle of a rubber sheet? Then you simulate Earth by rolling a golf ball or ball bearing across the surface?
It's important to understand what this is saying - the golf ball has no idea the shotput is there. All it knows is that the surface its running on is curved, and it's rolling along the path of least resistance on that surface.
Likewise, the Earth is not responding to the Sun, and for the most part is unaware it exists. It is simply travelling through 4D space along the path of least resistance. The Sun is not pulling on the Earth, it is pulling on space itself (spacetime) and creating the curvature.
The end result looks the same ultimately, but has a very different conceptual basis. Newton could never explain how gravity worked because he did see it as two objects pulling on each other, and there was no explanation of how one knew about the other. Field theories solve this problem - they're not pulling on each other, they're simply reacting to the shape of the field.
Now changes to that curvature do indeed travel at the speed of light, but that's not something that happens in this example - the Sun isn't jumping around a whole lot. So ultimately the field 8 minutes ago and right now are the same, or close enough for horseshoes anyway.