Both A and B are correct but B is an older form and is less commonly used now - you are more likely to see it in older literature and more likely to hear it in the negative form in phrases like "You needn't worry" "Well you needn't think I'm going to do it for you!" No "Need you a new hat?" is an archaic form - you can't say it now (other than perhaps in a few strong local dialects). "Need you [do something]?" still occurs as in your example "Need you go?" but never "Need you [a thing]?" Generally I would say don't write or use the B form in speech but understand it for the occasions you will come across it, mainly in literature or period drama. Hope that helps :)

A is the normal way to use 'need' with the normal meaning. People also say 'You needn't bother!' or something but that's a different nuance, like 'forget it, don't waste your time!'

Your B sentences were technically correct but very old fashioned (pre 1900s). Need you a new hat just sounds wrong, though I suppose it's a similar sentence structure.... My advice is to use subject + verb construction only with need. No inverted phrasing.

And what about phrases like "Need you a new hat"? May I say so or should I say "Do you need a new hat"?