I've been browsing some authors' articles and notice that several have downloads listed but still offer all rights. After something sells for usage, I understand we can't sell unique or full rights, so should we edit the rights offered, or will CC sort that out?
No, you don't have to edit them. Customers who buy articles that are offered only on Constant Content can see that downloads have been made already. However, they still have the option of purchasing for other rights, therefore removing the article and preventing future purchases by others.