It may be obvious, but I firmly think that Doflamingo's sunglasses are his "Straw hat", maybe there's more meaning in him wearing some sunglasses than in what lies behind them.
We have seen how much tenryuubitos are disgusted by the world, they even wear spacesuits to avoid to be polluted by what's unholy, even if they are going to visit regular nobles.
The most arrogant ones must think that common citizens doesn't even deserve to watch them in the eyes, And young Doffy was a really arrogant Tenryuubito.
So that's why he started to wear them, when he wasn't in Mariejois.
Also, remember that what the angry mob did to him still gives him nightmares?
Maybe Doflamingo suffers from Posttraumatic stress disorder more than it was hinted by his dream, and mimicking what a Tenryuubito would do, he wears sunglasses to shield himself from the world that he despises so much.
Behind the sunglasses:
Maybe he has just the same glare as young Law, only more determined, but looking tired and disenchanted. He wants to destroy the world, but he doesn't care anymore about judging how corrupted the world is from his point of view. He wants only to get his revenge, quickly as possible, he wants to destroy all, even the Tenryuubitos that prevented his return in heaven.
He's a lot like Lucifer with PTSD, if you think about it.
