I’ve worked with some pretty rotten software, but management software is easily the most user unfriendly, so my vote goes to HPSM.
There’s a bit of configuration for the channel list that you can do to keep what you want where you want. Sounds like you have a section set to only show unread, that’s a setting. Also, there are back and forward keys (and shortcuts for them too) to move between a series of chats like a browser.