One way we deal with this on other forums is there is a stickied thread at the very top of the classified section. Once items are sold the thread title or link is posted to that sticky by the OP and asks for deletion. A mod then reads it, deletes the sale thread and reports "Done" o "Got it". It does keep the classifieds section clutter free.
HOWEVER, It does require more work of the mods. This system works better on forums where there are many mods available. If that is not the case here, that may not be an idea to try.
It would be great though, if on current posts if the OP would go in and delete the items from the post if it's sold instead of just posting "SOLD" next to the item. That would keep the search engine from picking up the keywords in the post.