sian, there is a current sitcom in the US that features a female lead character named "Dylan"  (or "Dillon" -- I haven't seen it spelled).

Another relatively common man's name (nickname) in the UK that is almost exclusively female in the US is "Laurie," (although in Louisa May Alcott's time,  I guess, it wasn't all that unusual a nickname for "Laurence" in the US).

Funny thing:  I heard young folks on a talk show not long ago referring to "Ashley" as an exclusively female (or at least decidedly feminine sounding) name, apparently unaware that that was the given name of the object of Scarlett O'Hara's longing in "Gone With The Wind"  (played, interestingly enough, given this thread, by Leslie Howard).

McGrath, the male receptionist at a convalescent home in Seattle is named "Brooklyn."

I usually think of "Gene" as a man's name, short for "Eugene," but then there was the Hollywood actress "Gene Tierney" in the '40s and '50s.

And what about "Sidney" or "Sydney?"  Tom Sawyer's cousin(?) was a boy named "Sid(ney" and there have been many well known male Sidneys (Poitier, Greenstreet, Vicious, etc.), but is "Sydney" usually used for females?

"Jordan" is a pretty popular boy's name in the US these days, but there is a woman, about 90 years old,  in one of the nursing homes where I play music, whose given name is "Jordan," too.

Kat, we had a famous male "Connie" here in the US a while back: Connie Mack (Cornelius MacGillicuddy, IIRC).

And Robyn, I know a lot of females named "Robin," too.  (My brothers, FWIW, are Robin and Terry [NOT "Terrence"], and both have gotten volumes of junk mail or even business mail addressed to them as "Miss."

Other cross-gender names, historically, at least:  Hilary (or Hillary) and Carol (Carroll) [which is, I think, a variant on "Karl," "Charles," "Carlos," "Caryl," etc.)