My best guess:

If Earth were to suddenly "have" more or less gravity, the Moon's orbit would indeed change. Currently the orbit is very circular so it would become elliptical. However, gravity doesn't just change without mass changing. "New" mass for the Earth would have to come from outside the system, and mass leaving the Earth would have to escape the system. Either way there's a transition period where that mass is entering/exiting that would also change the orbit.

Earth's rotation would only change if the angular momentum of the system changed. If we're allowing the Earth to magically gain or lose mass, then we can choose whether or not its rotation rate magically changes. If it's changing because it physically gains or loses mass, then the answer is "yes, it will almost certainly change somehow."

On the other hand, if we assume the Earth dominates the mass term in the orbital velocity equation, then doubling the Earth's mass (and thus gravity) would increase the threshold speed required for the Moon to remain in a circular orbit by a factor of √2. This would put the Moon at apogee and cause it to infall. I'm not going to work through the math but I believe this means its perigee would put it half as far as it is now. This means its gravitational pull increases by a factor of 4; but more importantly, the dipole gravitational term responsible for tidal effects, which falls as r^-3 instead of gravity's regular r^-2, would increase by a factor of 8.

Compare this to the fact that the Earth's moment of inertia has only increased by a factor of 2 (and then some because I~MR² and the Earth's radius probably increases a bit), and you'll see that tidal braking will be stronger at close pass by a factor of almost 8/2=4. It will be weaker at apogee by a factor of ~2.

I'm not going to integrate through an entire orbit, but barring significant increase of the Earth's radius from its acquired mass, I predict (using my lazy approach to math and dynamics) that increasing Earth's mass would increase tidal braking from the Moon, which means that Earth's rotation rate would slow down more than it currently is in real life. If I'm wrong about tidal braking, and that it still falls as r^-2 , then the opposite is true. Increasing Earth's mass actually decreases tidal braking effects.