I haven't watched Toyko Godfathers but i will try and give you my recommendations for mature anime of these type. They are all going to japanese animated cartoons (i don't know many american cartoons that actually focus on drama and romance? americans can only do shows like the simpsons and south park, you shouldn't be looking there in the first place.)
Kimi ga nozomu eien - http://anidb.net/perl-bin/animedb.pl?show=anime&aid=896
just one thing to note, don't judge it by the first episode, it becomes a dark romance/drama after the first episode or 2.
(also not sure about the handsome guy part, I'm a male so don't know if i'm the best judge of that.)
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Lovely Complex - http://anidb.net/perl-bin/animedb.pl?show=anime&aid=5053
This is kind of goofy as it's a comedy, but i found it to be quite dark at times. The female and male protaganist hate each other with a passion at the begnning, but their relationship starts to develop more positively through the course of the show.
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Paradise Kiss - http://animenfo.com/animetitle,1806,rvrigz,paradise_kiss.html
This is a great romance series. It doesn't exactly have the girl hates boy at first scenario, i mean she doesn't like him very much at first, but she doesn't completely despise him. It's very mature and the story is great.
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Suzuka - http://animenfo.com/animetitle,1705,qukfrg,suzuka.html
I'm kind of mixed on recommending this one, It's good but not great. It has all the elements you're looking for. (except the handsome guy thing, you will have to be the judge of that) The main female lead hates our protaganist with a passion at first,and that hate remains pretty strong throughout the series. It does end on a good note though.
Worth a look
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The girl who leapt through time - http://anidb.net/perl-bin/animedb.pl?show=anime&aid=4151
Lastly, The girl who leapt through time is an exceptional movie. My favorite romance/comedy/drama film. Now it doesn't exactly fit your criteria but i'll recommend it anyways since it's such a great film. There is a good amount of drama, romance and comedy, so it doesn't feel to goofy. It may even make you cry by the end.
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Hope that is of some help.