Did every X-15 pilot become weightless when flying into the upper strato- and mesosphere?
Yes they did. Take away both rocket thrust and wing lift, and the plane goes into free fall. In this condition everything on board becomes weightless.
The pilot would feel only the light bursts of thrust from the small manoeuvring rockets, and the slow return of aerodynamic lift during re-entry.
You don't even need to go into outer space; check out the vomit comets used to train astronauts.