How is the elephant in "Shooting an elephant" the answer to Orwell's personal troubles?
Shooting the elephant is not the answer to Orwell's personal problems at all. He says he shoots the elephant because he is expected to (a sahib has to act like a sahib) and because he does not feel he--or any other white man for that matter--would have the option of doing nothing. He does it because he feels he must and because if he didn't, the crowd would laugh at him.
His personal problems don't go away when he shoots the elephant. He is still hated by large numbers of people, and he still experiences a value conflict between his function and his beliefs (even before shooting the elephant, he says he hated the imperialists, of which he is one). What changes for him is that he says he has a better understanding of the real nature of imperialism and the way despotic governments function: they function as he did, fiercely trying to hold onto their position and power by performing cruel, inhumane and unnecessary acts simply to avoid looking foolish and to retain their position and power.