I'm not particularly involved with the LGBT society at my uni as it's geared more towards supporting people who are still coming to terms with their sexuality etc., but I remember in the first meeting they said that the meetings were a "safe area" and they wouldn't make it public that you'd attended the meeting - so that if you weren't out no one would know you'd gone. So you could go to the society without "outing" yourself to everyone.
One thing I don't like is that the societies are generally called LGBT, whereas a lot of people are identifying as something outside of this - such as yourself. I'm also asexual, so I know what you mean about people not believing asexuality is real. People in LGBT societies are generally very accepting of sexualities other than gay / lesbian / bisexual - I really don't think it's something you need to worry about. If you do feel uncomfortable, you can talk to the committee members about this.
I think it would be a good idea to at least go to the initial LGBT meeting. As you're not out I think they could give you some advice / support, and you might meet other people like you.
LGBT societies have socials like other societies - pubs, scene crawls, that sort of thing. Mine also has talks and a "queer cafe" where you can drop in to get advice or just to chat.