If the moon wasn't drifting away and was in a static orbit, would the Earth's rotation slow and the moon's orbit speed up to the point where they're in sync?
The moon cannot orbit at a higher speed while keeping the same semi-major axis (average distance to the orbital centre of mass).
If you suddenly doubled the orbital speed of the moon right now, the apoapsis (the highest point in its orbit relative to the orbital centre of mass) would increase significantly.
If you slowly accelerate the moon in the direction of its orbital velocity consistently over a long time period, the moon will slow down relative to the Earth, but it's semi major axis will increase.
Actual scientists of HN: have at me.
Knowing what I know about that game I read this as: Warning; I actually know what I'm talking about here.
“Will the Moon ever leave the Earth’s orbit?” => https://youtu.be/IM_euz9PUiw
This is called tidal locking, and if the universe consisted of only the Earth and moon, this would in fact happen. However, the big heavy Sun also affects both the Earth and moon's rotations.
So why do I say that it has happened? Because the moon, having significantly less mass than the Earth, is almost tidally locked to the Earth. That's why we always see the same side of the moon. So the Earth's rotation hasn't synchronized with the moon's revolution, but the moon's rotation has nearly synchronized with the Earth's revolution (actually both the Earth and moon revolve around their common barycenter).
I don't know offhand whether that would happen before the Moon drifts too far away to remain in Earth's orbit, however.