What belief of George Orwell's is shown in both "Shooting an Elephant" and 1984?
The only real connection I see between these two works is the idea that people should act according to their consciences.
In general, "Elephant" is an essay about imperialism. But one of its critiques of imperialism is that imperialism makes people act in ways that they think is wrong. Orwell did not want to shoot the elephant, but he felt he had to.
In 1984, Winston is fighting for the right to act according to his own conscience. His society has taken that right away from him and he wants to get it back.
Does that sound plausible to you?