Clearly haven't read through all 6 pages of old posts, but for me, as apparently for others, the problem was related to the well known bug of the paper clip icon displaying for messages with attachments. Credit is not to me, but I'm too lazy right now to look for the original solution, so reproduced here without credit (and doesn't involve moving messages):
1) Create a rule where <Any attachment name> <contains> <"."> [include the quotes] and have the rule's only action be <Mark as Read>;
2) Apply the rule to the messages in question (Message > Apply Rules);
3) Remove attachments.