I'm a tutor on some courses and until recently I had a quick fix as to how I could subscribe to all forums without having to post on them individually or click "subscribe", and this really made it easy for me to follow my students' progress without having to click everything individually. Recently this site underwent some cosmetic changes and I can't find how to do this? Can you suggest anything?

I've been using moodle for a long time, but I haven't seen the opportunity to sign up for a course as a tutor yet. Are you sure you have correctly explained the problem you are facing?
Maybe you just made a mistake. Now there are many other resources for tutors. For example, I'm always looking for students on 上門補習. There is a very convenient system of communication with the client.
On moodle, there is only an opportunity to enroll in a course or create a class as a teacher. The moderator can add you as a second teacher to the system to make edits to the material.