@AidenWeaver -
I just confirmed on firefox (I am assuming that chrome is the same way) that as an instructor I can rearrange my dashboard, but as a student I cannot. I use chrome, but I have several scripts enabled that I did not want to turn off and try in student mode in chrome - which is why I tried firefox.
according to this guide page on the student dashboard, you should be able to drag and drop the classes.
I tried the move option (click on the ellipses for a course and select move) - that did not work in student view either. So from the student guide pages, it says you can move the courses on the dashboard, but operation wise it is not working currently.
My suggestion is to contact your Canvas admins at your school and list everything you have tried in rearranging your dashboard. Ask them if they can duplicate the issue and if so if they would contact Canvas support.
It sounds like there is now an issue with where I am assuming there wasn't one for you earlier.