A bit of a nitpick:
Astronauts aren't weightless because they're in free fall/orbit/space.
They're weightless because they're not experiencing thrust.
A famous thought experiment has a person in an elevator car in deep space. They can't see outside to judge their position, but they have all of their senses otherwise.
When that elevator is pulled "up" at a constant acceleration of 9.8m/s/s (that is, every second it's going 9.8m/s faster than it was before), this is completely indistinguishable from sitting still on the surface of Earth.
If whatever force was moving the elevator were to stop working -- that is, the elevator car is left to drift through deep space at whatever speed it wants to -- the person inside is traveling at the same speed as the elevator car.
If three skydivers were falling from an airplane, holding hands, and one lets go of the others, that one doesn't suddenly plummet to the ground any faster than the other two. They aren't "dropped," relative to the rest of their party, because gravity is working on everyone equally.
Since our skydivers are going the same speed as each other, they can push off and float around each other. The same for our passenger in our thought experiment elevator -- The elevator is drifting through space at great speeds relative to something out there, but since the passenger is going the same speed as the elevator, they're able to push off and float around, as if there were no gravity affecting them. Any gravity outside, such as from passing near a moon, goes completely unnoticed by the passenger, since the gravity affects the elevator the same as the passenger.
So, orbiting in the ISS, streaking through the rarest stretches of the upper atmosphere at a blindingly fast speed of 27,600km/h (17,100mph), feeling 90% of the same gravity that people on the surface feel -- they experience weightlessness because they're traveling at the same speed as their ship.
When the ISS uses engines to boost its orbit (because in Low Earth Orbit, there's enough drag to slow the craft down and bring it dangerously close to thicker parts of the atmosphere), the astronauts do experience some weight once again. As soon as the engines are off, they're back to being weightless.
Rounding back to your question, of astronauts traveling between different bodies such as entering orbit around the moon or Mars: No, passengers will not feel a difference in gravity, because any shift in gravity affects the ship and passengers in the exact same way.
If the ship fired its thrusters, or entered an atmosphere and aerobraked, the passengers would feel that as gravity -- but just going to different gravitational gradients does not change weightlessness.