I wouldn't say I "feel male", so like you (and a previous poster who responded, sorry I haven't read everything yet), I don't really believe in "essences" or "souls" or anything like that.
I don't know what male feels like. I don't think all women feel the same. Yet... Something around the "I", my identity, always took the form of a male in my imagination, if that makes sense. So, my body felt wrong, and I desired to look male, and imagined myself as male consistently.
I would have dreams and the "me" in them would be a guy, when I would read stories I would see "me" as the guy, I would play pretend as a guy, I would daydream and the "I" in those daydreams was a guy. I desperately wanted a male body. I wanted clothes to fit my body like a male body.
I don't particularly want to be "treated like a man" in the social way, I don't much behave like one anyway I suppose having been socialised as a girl, but I wanted to look like one.
A deeper voice, a more masculine facial structure, a male fat distribution, male looking legs and hips, no breasts, etc etc. I loved it when I would get sick and my voice would get lower... I even found myself (and I know this isn't right) wishing to get/be at risk of breast cancer so that I could "have a valid reason to remove them".
I tried just to live with my body until my mid/late 20s. Tried therapy and medications to help with how I felt and to just get on with it. It didn't go away.
So, I suppose in the end I'm not even saying I have a gender identity or that I'm a guy or have an essence. It's just that the only thing that relieved how I felt in the end was looking more male. I feel better using a male name etc. Could I walk around like this and just keep calling myself a woman? Yes, I suppose so - but that is also confusing for people, isn't it at times? If you really were to ask, I would agree I'm not a man, I'm a trans man, a woman who strongly desires to live in a masculine body?
Personally I try to use unisex facilities wherever possible.
Lots of trans people are open about being trans but I do think some hide it "go stealth" due to the transphobia - fear of being harassed and ridiculed and attacked. A desire to blend in and not be bothered I suppose. And many do go about like that, but they also aren't the loud ones you see in the media, and I'd argue they are the ones more likely to pass fairly well anyway... Of course that's not women's problem, I'm not saying it it, it is a problem though. I think the simple acknowledgment of their own sex is distressing for some of them, which I can empathise with, but also that's simply the reality of our lives.
Although, I myself would likely be labelled as transphobic by a large proportion of the community these days, as I don't think we literally are men/women, and I do think dysphoria is needed. It's a rare medical/mental problem, for which transition may be the best course of action for some (imo).
It is interesting to consider the transsexual/transgender split as well. I feel "transsexual" had such negative connotations that there was a desire to move away from that association...