Like Insideontheoutside, I don't like my moobs either. They're just annoying for me and they've gotten less and less appealing on any level now that I've got a hairy chest going on. I do indeed plan on top surgery, however, I'm worried that I'll never have the money, so I might just have to bind for life.
I don't think it's strange that you like having them. If they don't bother you, then that's a good thing, isn't it?
Unfortunately though, even if you don't see anything wrong in identifying as a boy and having boobs, I'm not gonna lie, the world will have a problem with it. People assume that all humans with boobs are women, "If it smells like it, looks like it, sounds like it, then it's gotta be a...." Sadly enough, chances are that if you keep the boobs and you don't bind or hide them, then you're going to constantly be addressed as a girl. You're not going to change the opinion of the whole world, not unless you become a pop-star, so instead, you're going to have to change the way that you think in order to be happy. Honestly, through the years, I've realized that I just have to stop giving a ->-bleeped-<- about what everyone else says or thinks about me in order to be happy, if I cared, I would have hung myself years ago.
I do however think it's strange that you're saying that binding has ruined your back and such. Do you mind me asking how exactly you bind and for how long you've been doing it? Perhaps you should try another method? Binding should never hurt and when it does, you need to find another way to do it and give your chest a break.