I'd be willing to bet it's exactly that - they feel like "Doug" died. What they aren't seeing is that "Doug" was never, really, truly "Doug". Who he became - who you are now - is who he really was the whole time. And in a way, he -did- die. He sacrificed himself so that Dodie - you - could be...born, in a sense. But that's not something that most people will ever be able to see, or even if they do see it, they won't be able to understand it.
That's transitioning. In becoming who we truly are, the person we presented as beforehand must perish. We're phoenixes, really - that identity, that physical shell, must burn away, revealing the...truth beneath the lie, so to speak. And they may feel like you're responsible for the death of the person they knew, as though you walked up to him, shot him in the head, and took over his life without them knowing, and now that you're truly yourself, it's like you'd been wearing his face as a mask, which in all technicality you have. They may feel betrayed. I know that when I came out, several of my friends told me they "didn't know me anymore" and that "had they known when they met me, things would have been different"...different in the sense that they would not have bothered befriending me.
Ultimately, one true fact remains - you are a beautiful phoenix, risen from the ashes of a former life. You have been reborn. And how many people can say that, in the sense that they are literally a new person?