I am a bit late to the thread, but - if you have any kind of hay in the cage / handle hay, that could also a be a likely cause.
I have never been allergic to anything as far as I know and also developed an inexplicable, highly unpleasant rash on my lower arms several months ago. My dermatologist was quick to blame it on the hamsters, but - since I've never had any reaction to anything or any animal and this was the first time it had ever happened - I didn't entirely believe him.
From what I read up on then, hay or, as has been suggested, substrate are likelier causes. (It later turned out it was something else altogether and no allergic reaction to anything.)
I also would have kept my hammies no matter what. One suggestion in the case of allergies was: long sleeves and, if absolutely necessary, gloves. And making sure that you wash your hands thoroughly after handling whatever could cause the symptoms and before you touch anything else, particularly your face.
From what I've read, there are many people who successfully keep hamsters despite (proven) allergies to them. It seems that you can easily make the necessary adjustments in most cases, and that they often don't even involve medication.
Wishing you the best.