are students fully exempt from council tax?

I graduated this summer and moved out from my private accommodation in June (contract ended in July). I received a letter saying that I need to pay tax for the tenancy between April-july (exemption for students and discounts included). I'm confused because I thought students were fully exempt from tax? Am I just being stupid?

Just take your enrolment letter which shows the start/end date of your course and they will waive the charges applied.

The end of my course date is the 28th June, whereas the end of my tenancy date is 26th July. and it seems like they've charged me for the period between my move out date (5th june) and the tenancy end date. Is this expected?

They should charge you for 29/06 to 26/07 nothing more.
Did you inform them that your course has ended on 05/06 and you are moving out?

they only asked me what date I moved out, which was the 5th June. I assume they thought that was also the end of my course date?

Yes I think you need to visit the council with your course dates letter and sort it out.
It’s pretty easy process and even can be done via emails if visiting is an issue.

It's likely that the council contract out the work of collecting council tax to a third-party company; so there probably won't be anyone at the council itself dealing with this.
I'd expect you can sort this out via email.
It's likely that your student status for council tax purposes will end at the end of your university's summer term, and that you'll be liable for council tax from the following day up to the end of your tenancy. It sounds like they've incorrectly interpreted your moving-out date as being the date you stopped being a student.

does it matter that I was a final year student? On my degree transcript, it said the course ended around 26th June. So will I have to pay for the remaining 1 month?

You’ll need to pay for the period that you weren’t a student.

Just to note, part-time students are not exempt from council tax in any respect! Also every person in the household needs to be a full time student to get the council tax exemption - if you have a mixed household with some full time students and some working professionals or part-time students for example, you wouldn't be exempt, at least that's been my experience.
This is more just to clarify in terms of the thread title, for posterity. You seem to have had good answers for your specific situation already in this thread