I was charged the cancellation fee four times

I would pick up the phone during normal business hours M-F and ask to speak with a billing supervisor.  Support phone numbers vary by region.  Google it.  And best of luck!

Contact your bank and or CC company and dispute the charges.

First, let me move this to the Get Started forum, which is the appropriate forum for issues like this. The Community Help forum is for help in using the Adobe Support Community forums.

Second, I'm not sure what to tell you. It sucks that Adobe customer service and your school's IT department are pointing fingers at each other. You can try the main Adobe customer service portal here:

https://helpx.adobe.com/contact/support.html

You must be signed in with your Adobe ID and allow cookies in your web browser for this to work.

This is the link to the US website. If you are not in the US, the site should redirect to the page appropriate to the region associated with your ID.

Click the chat icon at the bottom right of the page to open a chat session. Typing AGENT into the chat window should connect you directly to a CSR rather than the AI.

If the CSR refers you to the forums, insist on speaking with a supervisor and keep moving up the chain until you get someone who knows what they are talking about.

You can also search for Adobe customer support using your favorite search engine. This will normally provide you with a regionally appropriate phone number that you can use.

Lastly, you can probably call your credit card and dispute the charges. That might actually be your most effective tactic. It's odd that each charge was on a different day. Did you try to cancel it multiple times?