The answer to the question, "Do the astronauts feel the station moving?" is yes, definitely, but sometimes in an "indirect" fashion.
During Space Shuttle mission STS-109, when floating in my sleeping bag and waiting for slumber to come, I would notice that occasionally my body would softly brush up against one side or the other of said sleeping bag. A quick glance at a laptop computer nearby, with, among other things, thruster status on its screen, would validate that a vernier jet (a small, 24 lb thrust attitude control jet) had just fired, in response to the autopilot, which was keeping the Orbiter in a certainits previously-commanded attitude.
So, really, it was not a case of my body brushing up against the side of my sleeping bag, but a case of said sleeping bag (which was attached to the Orbiter) brushing up against my stationary body.
A measly little 24 lb thruster pushing around a 220,000 lb. spaceship.
Cool stuff!