In what sense has the Doctor been running from the Question "all his life"?

It seems to me that the question and the answer given to it at Trenazalore are being confused here. It is true that that Doctor's true name was used to open the Tardis, but the question, at least as it was understood by Dorium was "Doctor Who?" The Doctor wasn't running from the Trenzalore answer, because clearly he already knew his own name. He was running from the question. Although in some sense, his name is his identity. The special precautions he takes not to reveal it appear to be tied to his heritage as a Timelord.

The story of the Doctor is grounded in his shirking the responsibilities of life within Gallifreian society to travel the universe, stealing a time machine to do so. In doing so, he rejects his heritage, and starts running. He becomes involved in the problems of countless other species within the galaxy, and chooses an itinerant life.

What Dorium is therefore saying when he says that the Doctor has been running from the question all of his life is that the the Doctor's constant and relentless forward momentum without an overriding objective is intended to avoid the Doctor having to come to terms with his own self and his place within the universe. The answer that the Doctor is seeking to avoid is the answer to the broader question of who he is, rather than the narrow answer of his true name.

Of course, this may simply be Dorium's own perspective on the Doctor's actions. It is quite possible to interpret the Doctor's life path as having come about because he had a great deal of self-knowledge about who he was, and more importantly about who he wasn't. When faced with a choice between life on Gallifrey and escape, perhaps the Doctor already knows that he cannot be himself in the former.

From another perspective, it's a question that's never really answered anyway. Our experiences change us all, and who we are is a process, not a destination. In that sense "who are you" is a question every sentient being spends their entire life running from, because self-discovery is an inherently painful process. But now we're straying into the realms of philosophy.