Let's go through this again:
Does DNA determine sex? Yes, your genetic code contains the instructions for your sexual development. XY is male and XX is female.
Are sexual organs part of what makes someone their sex? Yes, a MTF transsexual has a penis that has been surgically altered to look like a vagina, they do not have a vagina nor do they have female organs (uterus, ovaries, etc). A FTM transsexual still has a vagina that has been surgically altered to look like a penis, they do not have a penis nor do they have male sexual organs (prostate, testicles, etc). In both cases the natural sexual organs have been altered by surgery to appear different, but they are imitations of what is real as far as the opposite sex goes.