Genuinely if you had asked me whether I felt like a man I would have said to you "I don't know. First define 'man'". As far as I am aware, I just 'am'...
For me to feel "uncomfortable" with my gender would be like feeling uncomfortable with my national insurance number.
This is hardly surprising, given your own internal image of your gender (whether or not at the forefront of your mind most of the time) has always coincided with the image imposed upon you from the outside.
People like you and me, who don't experience any dissonance in this regard, have little reason to be particularly aware of their own gender most of the time (I'm rarely consciously aware of it either).
What if you woke up tomorrow and found that everyone, from your family members to your colleagues, were referring to you as "she", expressing surprise, amusement or disapproval (not necessarily overtly or deliberately) at your choice to cross-dress in men's clothing, etc. (and there are many more etceteras).
Would you feel equally as comfortable in those circumstances?
Would you be no more aware of your gender than you are now?
And anyway, putting all that to one side, does the question of how society should deal with transgender people really turn in your mind on how you or I should happen to feel about ourselves?