Hey Salad! I just wanted to chime in and say I'm using my school's Disability Center for the first time this semester. I normally work directly with professors. However, I had all new profs this semester and my symptoms have been intense the past few months, so I wasn't as comfortable approaching them this time.
The Disability Center only needed a letter from my therapist. From there, they contacted my professors for me to request certain accommodations (late assignments, leaving in the middle of class, absences without doctor's notes, getting to sit in a place where I don't feel trapped, etc.). They only shared what I wanted them to. From there, I felt more able to speak to my professors directly. I think having the Disability Center gave me more credibility and made my profs more inclined to work with me.
Also, kuddos for calling! I totally understand the shame/failure/humiliation fears, and I think it's awesome that you reached out despite those. I know just acting to help myself when I feel those things is a struggle, and I'm really encouraged by you.