Ok, I think I've got a solution in place now that should help.

A member's first post/posts will be hidden from the public and need to be manually reviewed by the mods. If approved (valid member), then that member can post freely. If deleted (spam or other some other form of trouble), then management can simply ban them.
And I also removed about thirty registered but yet-to-post spam accounts, and patched some security issues in the third party forum software that I think allowed this to happen in the first place.
So, we'll see how things go in the next few weeks, but I'm pretty sure that things should be better on the spam front!