"I like it when people sing in character and away from their own gender too, there's something very folk-song about it."
So do I. Off the top of my head, the only example I can come up with is Emmylou Harris
doing The Boxer. A less clear-cut example (since the singer does not refer to herself as a man) is "Valerie," a song done
amazingly well in
this video by Amy Winehouse. Do you have any others in mind?
(PS -- I can't think of any in which a man sings a song that is meant to be a woman speaking. The closest I can think of is John Prine's "Angel from Montgomery," which, come to think of it, certainly qualifies, but since he wrote it, and it was intended to be the voice of a woman, it's more a persona song when he does it, though perhaps even more powerful when Bonnie Raitt does it).