No, the earth must be moving at the same rate as the sun in order to even be in orbit. So there is no danger of 'slippage' of that orbit due to the sun's movement over time. Remember that (as far as we know) the earth formed from the same cloud of gas and dust as the sun. It has had the same basic motion/momentum from the beginning. And we orbit the center of mass of the whole solar system, not the sun itself. (Yes, that center is probably about 6 centimeters from the center of the sun...
)While it's true that they have not yet figured out how to
prove that gravity propagates at the speed of light, your supposed 'insight' is a misunderstanding of orbital mechanics, not any sort of proof that it propagates faster than light.