We've just added a new "Ignore" feature that can be used for both forums and threads. To ignore a forum or thread, go into it and click the "Ignore" link at the top like shown below: Ignoring a forum means you won't see topics from it in your new post list. So this effectively gives you the same functionality you had with the old Combined Section list. You also have the option of ignoring a specific thread which we never had before.
With the old Combined Section, if I didn't choose to only show new posts, I could continue to see all the recent posts including those I had clicked on. Is there a way to make that happen with this new format?
Yes, there is. When you're looking at the New Posts page, on the right is a link to show all recent posts. I've highlighted it below: