Can you prove that NASA used a different formula?
Your suggestion is that tens of thousands of civilians who worked on this project, one of the most advanced technical feats of the generation, found proof that Newton was wrong and kept it secret?
What about when the Soviets landed drones on the moon, did they find out the other formula and also keep it secret? Why would the Soviet Union not told the world of their massive discovery that reshapes how we view the universe! They loved proving how scientifically advanced they were, so did the US to be honest. Why would these cold war enemies work together to keep this secret?
What about all the other nations that have gone to the moon since? A total of 25 nations besides the US and Soviet Union (which is itself now multiple nations) have sent missions to the moon. Why would the thousands upon thousands of scientists keep this secret? If they reached the moon calculating the moon's gravitational field as anything but f=GmM/d2, why would they not show their findings? They just disproved Newton, the father of modern Physics! Not one of the people were willing to show their evidence?
Because right here are the numbers for landing on the moon using f=ma and they conform to all measurments of the landing.
It's not my job to prove the existence of Russell's teapot. If you're going to rest your claim on something that requires this much conspiracy, it's your job to give any explanation as to why and how.
We can observe with a telescope the trajectory of planets and heavenly bodies. How do we know the exact time for the return of Halley's comet? Because the universe works entirely and universally on F=ma. Because for centuries and confirmed by millions of observers over those centuries, every observation confirms a universal force of gravity that does not make a distinction between the earth and the rest of the known universe.
Do you honestly believe that these millions of observers falsifying the data? All of them?