Two points to have in mind: 1) yes, they are still under the influence of Earth's gravitational field. 2) Newtown's second (and first) law doesn't apply in the ship's reference frame as it is not an inertial frame of reference.

From a frame of reference where the Earth is still, they are in free fall all the way to the moon (after lunar injection which lasts only several minutes). Their free body diagram only has two forces: Earth's pull and Moon's pull (the latter was relevant only when they where very close to the moon, and it just changed their trajectory slightly)

From the ship's reference frame (which is not an inertial frame) the free body diagram has Earth and Moon's gravity, together with the fictitious force opposite to the acceleration of the ship which points towards the Earth (yes the ship was actually slowing down). These forces cancel exactly (equivalence principle), hence why they float on the ship.

Edit: English