I think you're right and wrong. Most people don't look at the individual areas, and thus, the posts die sooner than would in the general public area.
This isn't a mod problem, as they shouldn't have to be moving ANY of the posts, they should be posted in the right place in the first place. HOWEVER I think they should find a way of getting people to look in those areas more often. This is more of a user issue than a mod issue.
My suggestion is this.
Creat an "active" page that encompasses postings from every group, that way you can look at the active page and see everything that's pending or recent, and then if you want a specific inquiry, go to that section to browse them all.
I think everything should be in a designated area, even if one area is listed as "general posts", but that ACTIVE folder is where the general public would look. This would tend to solve both problems pretty well, with less "moving" to be done by the mods, and more attention to the more recent threads, no matter the area they were posted in.