No; no - it's a much milder effect than that in all the relevant ways. Basically we're going as fast as possible - light speed, straight in to the future, relative to nearby, stationary, objects. When you provide a velocity, you need to specify what observer is measuring it; that is something we see immediately from relativity.

Everything moves at the same speed, in a certain sense - "c", we call it. That seems counterintuitive, but it isn't, if along with the rotations you can picture there is another dimension in which one might steer - or rather, steer away from. We could take some of our speed through time and exchange it for some speed through space. This is time dilation. It is one piece more complex than that, as we are really talking about imaginary time, here. But "imaginary" in the complex plane sense. Flipping our perspective so that linear motion in one direction represents translation in complex time is called a Wick rotation.

You're thinking in terms of gravity, and I'm answering with geometry; but they are equivalent descriptions.

If you could sit somewhere near our observational horizon - say, 13 bly away, or somehow protected and hauling around a black hole, then we would appear to slow greatly - but also we would be very redshifted, and it would be very hard to get back. So practically you can kind of ignore it; but there is something there, if you really really want to run the world ahead into the future a little faster, and you are allergic to anesthetics.