Oh my god, my shoulder is throbbing and I don’t know why?!

Pinched nerve in the neck? Slept on that shoulder all night?
Beyond the scope of an anti-inflammatory otc pain medication?
A sling to carry the weight of the arm for a while may help, thus relieving the shoulder of that burden for a while?
Also poking to find the heart of the tenderness, and where to place the cold pack?
Also a bit of distracting? Like find something else to focus your mind on while waiting for the above to work.

You can still "teach", without putting too much stress on your body. Just explain to your class, I am sure they will understand.

Try a lidocaine patch.

If you can’t afford a few days off from paid leave work, you’re obviously not working in the right place

You need a gentle rub