Effete and effeminate are neither synonyms, nor shades of the same mannerism. One can be effeminate and effete, or masculine and effete.
He certainly isn't queeny, or nelly, or mincing- but he is effeminate (speaking from a personal knowledge of him when he lived in Los Angeles) and he is not effete.
I'd use the word sissy, but it seems to have fallen from favor.