During a transfer, the only force acting on us is gravity.
This gravity can indeed come from multiple sources, like the Earth, the Moon, the Sun, etc, all at the same time. In fact, there's never a time where these gravitational forces do not act on us, as the range of gravity is infinite.
So yes, astronauts are in freefall around both the Earth and the Moon. Freefall just means "the only force acting on us is gravity", so it's not relative to anything. You can't be in freefall around one object but not around another.
Feeling weightlessness can indeed happen even if there's a force acting on us (freefall is not equal to no gravity). We can actually not feel force! What we can feel is if there's a difference in the forces acting on different parts of our body.
When standing on the ground, we can't feel the gravity, because it's acting equally on all parts of our body. What we do feel is that the ground is acting on our feet but not the rest of the body.
When the rocket engine is burning, the engine is acting on our ship and not us. Therefore, there will be a difference in acceleration, and we will be pressed against the floor until this acceleration is equal. We feel the floor acting on a part of our body.
When in transfer, gravity affects our body and the ship equally, so there's no difference in acceleration, so we are not pressed against the floor, and are therefore feeling weightlessness.