Your intellect creates a world through which you experience the physical world. Your body is the bridge between the two providing sensory feedback through your brain to your mind and making your intentions into physical actions. My mind informs my physical experiences, not the other way around.
To my mind you have this exactly backwards. It's actually your physical experience of the world that informs how your spirit/intellect develops. There is no way to experience the world except through the body you have. For that reason, I do agree with the next part of your paragraph:
Obviously my body is female but my intellectual self is also female. I'm not a person inside a woman's body I am a woman inside a woman's body.
I agree and I think that some radical feminists do a disservice to women when they insist that our 'personhood' must be separate from our womanhood (which they see as something imposed on us).
However, we might be coming at this from different perspectives, Outy, because you seem to think (if I've understood correctly) that there is an extra-corporeal essence of femaleness (and presumably maleness) that has nothing to do with the body, and what I'm arguing is that it is through experiencing the world in a sexed body that our sense of ourselves as women (or men) arises.
This is why I disagree with the radical feminists who argue for 'gender abolition'. To me that's an impossible and indeed anti-human project, because the physical differences between the sexes will always be relevant to how we experience the world. We are a sexed species - there is no unsexed experience of 'personhood' separate from our male or female bodies. My embodiment as female is not incidental or irrelevant to who I am.
When I've stated this in a radical feminist circles, I've been accused of believing in 'lady brain'. But my argument has nothing to do with brain structures, and I actually find the modern obsession with locating all sense/feeling in the seat of the brain to be reductive and a faith-based belief, akin to the Ancient Greeks' notion of the four humours. Yes the brain generates consciousness, but our sense of our whole self is experienced through the body: through having limbs, a nervous system, a digestive system, etc.
Men's and women's experience of the world can never be the same, because our bodies are so different in so many crucial respects: not just reproductive organs, but hormones, bones, muscles - medicine is just starting to realise, for instance, that you can't treat men's and women's bodies as interchangeable for the purpose of diagnosing heart disease, or prescribing drugs.
This doesn't mean that I'm not for interrogating the cultural basis of femininity and masculinity - these phenomena arise, I believe, mainly from power relations between the sexes - they are not natural, except in the fact that the power relations themselves arise from biological differences in the first place. It's been stated here before that if women were as strong as men and had the same capacity for inflicting violence, patriarchy wouldn't exist. I believe it, and I also believe that what we understand as 'feminine' behaviours wouldn't exist either. However, given that these differences can't be abolished without getting rid of sexed bodies themselves, does that mean that the power relations will always exist to some degree, no matter what social structures we come up with to mitigate them? Yes, I'd say, as depressing as that is.
This is an excellent article that deals in part with this question:
Gender is socially constructed (upon material reality)