The sinking feeling is real in certain cases, even though it's not necessarily caused by the aircraft descending. On landing, the sinking feeling results from the actual descent of the aircraft towards the airport. However, there are other situations in which it might feel that the aircraft is sinking even though it isn't. Since the physics of the situation produce the feeling, there isn't any way to avoid it, although you can try to condition yourself to be less worried about it.
One example is a reduction in rate of climb after takeoff. When an aircraft takes off, the pilots direct it to climb very briskly until it reaches an altitude (usually 1000-1500 feet) that carries it clear of any potential obstacles such as terrain or buildings. Once it reaches that altitude, the pilots reduce the rate of climb and the aircraft climbs more gently towards its cruise altitude. When the pilots do this, it feels as if the airplane is sinking—but in fact it is still going up, only not as fast. It's similar to the feeling you get in a rising elevator as it slows down to stop at your chosen floor.
Another type you may feel a sinking sensation is in a turn. When the turn begins, because of the way coordinated turns are carried out, the airplane feels as if it is rising, rather than turning. At the end of the turn, it feels as if it is sinking, rather than straightening out. This results from the physics of turning aircraft, which lean into their turns like a bicycle or ice skater. In a turn it feels like the airplane is rising and then falling, but in fact it stays level throughout the turn.