So, as soon as star radius becomes smaller that Schwarzschild radius it becomes a black hole. But as matter continues to fall on it attracted by the gravitational pull its radius sooner or later will exceed Schwarzschild radius and it will become a visible heavy object again. This oscillation in/out of black hole state will continue. Is this a correct picture?
A better mental picture would be that, after the star's radius shrinks below the Schwarzscild radius, the matter behind the horizon undergoes a complete gravitational collapse, forming the singularity in the center. There is no force "strong enough" to stabilize the star and have it be hidden behind the event horizon. As far as we know, the singularity is "point like" with no radius. However, probably due to quantum gravity effects (which we don't really know about, this is just speculation) many people think it has a "radius" of roughly the Planck length, or something else very crazy happens entirely. Nobody knows. But according to general relativity, the singularity is a point.
If new matter enters the black hole's Schwarzschild radius, the radius expands, and the recently entered matter also goes on to fall into the singularity.