I'm a little surprised that the university doesn't have any dorm facilities with accessible features that work for you. I'd have expected that they would have developed policies for accommodating students with disabilities long before this year.
There's a lot of good information at swalsh's link in post #4 (http://www.disabilityrightsca.org/pubs/530901.pdf). Here's an excerpt:
I am attending a school that does provide housing options to its students. May I request reasonable accommodations with regard to my housing? Is the school required to provide me with an accessible housing option?
Yes. If your school provides housing options for any student, then it is required to provide both reasonable accommodations to its housing policy and accessible housing options. The school may not charge more for accessible housing or housing provided with a reasonable accommodation than it charges for comparable inaccessible housing. Examples of reasonable accommodation in campus housing include housing located close to campus or to a shuttle or bus stop, priority for single or larger units, or allowing a student to have a service or companion animal despite a "no pets" policy. The school must make accessible housing available in a number of housing options, so that students with disabilities have housing opportunities equal to those of students without disabilities.
At the end of that document there is a sample request form for the student and a sample verification form for a medical professional. You should read the whole document; it goes into detail about the kind of documentation you need.
I'm not aware of any case law on this specific issue, but there's a well-known case in which a university was charged with violating the Fair Housing Act by refusing to permit a service animal for a student with disabilities (US v Millikin University).
I'd recommend that you contact a disability advocacy group on campus or in your area. If you're not able to resolve this with the university, you can file a fair housing complaint online at http://portal.hud.gov/hudportal/HUD?src=/program_offices/fair_housing_equal_opp/online-complaint. They have an app now so you can file with a smartphone.