You were correct until the melting part. Sun reaches Earth nowadays too, but I don't see Antarctica turning into water in a matter of years.
The nuclear winter ending does not cause a flood, it just makes the conditions bareable. Allowing plantlife of the temperate climate to grow in most locations. However the sea level is stil very low, former seabed is a desert and the glaciers still cover all of Scandinavia, northern Russia, Siberia, Alaska and Canada.