I guess if you really want to stay as anonymous as possible, you could find a trustworthy person here on this forum and send them $20 in cash a month thru the mail to that person, and they could release you coins. You can buy coins from lots of people on this forum, including me. Just make sure you use a trusted escrow service.
$20 can get lost in the mail. If that happens, the sender will not know if the seller really did not receive it, or just took the money.
The seller has to give his address, so one party loses all privacy. I would not give my address just so that someone else can be anonymous.