Take a deep controlled breath. Seriously I hyperventilate when I get stressed out. Oxygenate the brain.

Ok, it seems to me most of what you are panicking over are completely out of your control such as the unis response to your appeal or the number of admittances.

Perhaps a better use of all that mental energy is to focus on only what you control, which is writing your appeal letter and taking care of yourself. You have absolute control of putting words in that page, printing it out, and getting it in on time. If you want to switch majors and discuss it, then do. You want to describe the difficulties with your illness, go for it. This is your chance to express yourself, so you can take it.

I would advise writing everything out first, just get it out. Then you can edit it down. Agonizing over word choice, grammar, etc., the first time through is painful and wastes time. After you edit it down, get someone you trust to be honest with you to read it over. The trust part is important, because more than periods in the right place, they can tell you what makes sense, what doesn't, what helps or hurts your case. This person needs to know you and your situation. You also must be willing to take their constructivism criticism.

I know this all feels like a huge, life altering moment.
Regardless of what happens with your appeal, though, you really will be ok. You sound determined, resourceful, and dedicated to getting an education. I promise as a person like that you really can.