I do hear what people are saying re confronting him and potentially not getting answers, but we need to bear in mind that this is not mainstream cheating we are talking about, the reality is that he may not admit to anything because he may not be ready to do so, even to himself.
If he does have issues re his sexuality then he is clearly not comfortable admitting those yet as he's supposedly in a heterosexual relationship, with a woman.
There are a lot of emotions at steak here, not just the op's, but her dp's, those of his family (I can't imagine it's easy as a parent potentially being given that kind of news about your son), his friends, work colleagues... etc.
I do believe that op needs to know, but in this instance I don't think that asking him is the way to go about finding out, because he may simply not be ready to tell. And it may have nothing to do with the wish to betray, it may simply have to do with the fact he can't admit it to himself, many people struggle with gender identity issues for years because it's still not seen as an accepted way to be, and coming out as transsexual means changing your whole identity, and the associated rejection that goes with that (and there is rejection, because not everyone is accepting). That is why so many men, and women transsexuals do marry, and have children, and do actually have close relationships with their partners before coming out.
Op - you do need to know, but I think that you unfortunately do need to go about it in a slightly more roundabout way, because I think it's unlikely he'll tell you.
So I would gather some evidence, ask him for a picture, or even just his name?
Only once you have this information can you begin to decide where you go from here.
Many people do stay with their transsexual partners, my friend would have if he hadn't decided to start seeing other men, but that is a very personal thing, and you should not feel pressured into going or staying.