Do what other easy-to-join sites do. Whatever that may be.
Site changes due to massive spam
It should appoint older members to flag spam. 3 incorrect markings a month or so, and you lose your ability. Flagged spam would be dealt with quickly.
Captcha is free. Just have to pay one of their programmers to put it in the forums.
It's a person making the acc, not a spambot, I think, so that wouldn't do any good. A spambot would post topics faster, I think.

What is chess.com going to do with the huge amounts of spam flooding the forums every day? Maybe making some changes in the sign up process so it's harder to create quick account for spamming?