So I have TERRIBLE allergies, and I can sympathize. I am allergic to almost all animal dander, about 12 different trees and grasses, hay, molds, dust, you name it. Not fun for an animal lover who rides horses.
My best advice would be ALLERGY SHOTS!!! Once they did the skin tests on me and found out what I was allergic to, they were able to fix up a serum and I went weekly to get the shots. They were itchy and a pain, but oh my goodness did they help. I used to get sinus infections 3-4 times a year, and since starting the allergy shots in 2007, I've had maybe three. They don't erase the symptoms entirely, but they help immensely. I have heard some people who say the allergy shots didn't help for them, but in my experience, they changed everything.
Respiratory symptoms are tough to deal with, the doctor can probably give him an inhaler to help. Prescription strength antihistamines would also help, especially if his allergies are considered minor, and shots seem unnecessary.
The other thing that might help, is not only to move all your hamsters out of the room, but to thoroughly clean the room -- dust, vacuum, wash the sheets, the whole nine yards. I don't know your apartment set up, but if possible, you could make an "allergen-free room" where he could be comfortably. At my parents house, the cats aren't allowed into my room, its my safe, allergen-free haven.
Hope this helps! I've found that as a pet lover with allergies, it IS manageable, it just takes some extra work. Just get him to the doctor soon