I appreciate the honesty in saying that you don't know what a woman is. But if we're going to reject the biological defnition of "adult human female," and not replace it with anything concrete, and still use the term "woman," then it still seems to me that we are dangerously close to claiming that womanhood itself is not objetively real.

But if you say it outright then the "TERFs" can accuse you of "erasing women." And possibly even worse, the claim that "trans women are women" doesn't really have the same weight as a truth claim.

The crux of the problem comes down to the fact that we're attempting to define man and woman by separating the concepts from biology. My question is, "what else is there?" Is there a soul in which one's true gender resides?

If you can permit me this observation, I think that is the actual argument that you are making. You are arguing for a gendered soul in which one's womanhood or manhood exists most truly within their consciousness. And of course, the soul cannot be defined, observed, or proven. It can only be felt, known, experienced, and believed.