Also want to support and suggest that MNHQ read the Guy Called Helen post.
'Woman', 'female' and 'she/her' all refer to people who were born female. 'Man', 'male' and 'him/his' all refer to people who were born male.
Yes, language moves on and develops spontaneously but that is not the same as deliberately imposing and policing changes to language which deny reality and eliminate the ability to describe it.
People don't feel male or female, they just are. People might feel masculine or feminine. Personally I can't find any use for those words for myself because I think the construct of gender is bullshit, but if you believe there are two genders with specific characteristics then you can go ahead and use masculine and feminine to describe yourself. Describe yourself as gender non-conforming if you like. I can see how you might find that helpful in a world where the construct of gender exists in some people's minds. Personally I think it risks further entrenching the idea of gender rather than challenging it, but I don't need everyone to agree with me on that point. However you describe yourself, as far as I'm concerned we're humans with a range of human characteristics.
You can call me a cis woman if you want, but it doesn't make any sense because you have no idea whether I identify as a gender at all.
If you prefer to present yourself as the opposite sex in terms of appearance (dress, hair, makeup) with or without hormones or surgery, then I absolutely support you in doing so and empathise with finding your body difficult to accept, but that doesn't mean you have become a man or a woman. You are a man presenting as a woman or vice versa, or a transman or transwoman. I don't think for a second that a transwoman is any less than me but the transwoman is not a woman.
The link to the equalities legislation was really helpful (and eye opening!). The example they give (a transwoman being served in the pub and being repeatedly called sir and him, despite complaining) describes a situation where there would have been no need to use any pronouns at all, so deliberately choosing to do so and using male ones, even after a complaint, probably would be an act of aggression and harassment and it's quite right that people should be legally protected from that (and from not being served etc). However, if you are discussing a third party, the use of third person pronouns is unavoidable without using vey clunky sentences. The law as described seems to apply to organisations and businesses who might employ a trans person or from which the trans person might receive a service. It isn't clear that it applies in the context of individuals having a public discussion.
Even if the law did apply in that context, it could still be a reasonable response to challenge the law by protesting against it. Pretty sure that's how we got the vote. Sometimes protest means ignoring a law - Rosa Parks for example.
Some people will prefer to use she to refer to some or all transwomen. Maybe they will make a distinction based on the degree of transition and maybe they won't. Some people will prefer to use he, as this matches the sex of the person being discussed. I think that's fine, even if the person is offended. I would also absolutely defend the right of others to offend me. It can't come down to what offends people because that is so subjective.
With regard to trying to use they/their in sentences where it just sounds awful, we actually do have a singular neuter third person pronoun - 'it'. We use it quite comfortably to refer to unborn babies when we don't know the sex. To use it for person when the sex is known would be seen as contemptuous but that doesn't have to remain the case - language could evolve to make the usage acceptable.