The method I was suggesting did not involve marking everything as read first. That would prevent it from working.
Your "Watched Threads" are sorted by the time of the most recent post. So if you start at the top of that list, find your last read post, read the rest, etc., that thread won't appear in your list at the top again until there is a new post in it, and then clicking on the blue button will take you to the new posts.
In my case I worked through all my watched threads that had new posts Monday, since the site came back up with the new software. So for any of those threads, clicking on the blue button works. The problem is the first time there is a new post in an old thread I'll have to find where I left off. It is this problem that would be avoided if there was an option to mark all posts in a group of threads as read. I have sent this suggestion to both Danny and Doug via a "Conversation", as Danny requested in another thread.