I'm having the same problem and it happens even when I use a browser that clears cash and cookies every time I close it. I cannot save or change bookmarks when using the scriptures through the web interface. I can occasionally add new bookmarks, but it doesn't work consistently. Oddly enough, I can delete bookmarks without a problem. I think church tech is like Microsoft; they keep adjusting things that work until they don't work anymore. Then they have to start over.
Brice