There are a lot of different medications that don't involve aspirin - different types of anti-inflammatories, as well as various disease modifying drugs. However, I'm not a doctor or pharmacist and given your allergy, it will be safest for you to consult with a rheumatologist about your options. While I'm at it and just because I have a similar type of allergic reactions to certain things myself, I feel compelled to ask if you're carrying an EpiPen, just in case? If you don't, get one - it can save your life.

I am concerned that your doctor is repeatedly stating that he "doesn't know" about alternatives to Tylenol in treating RA. My concern is about two things in particular. First, his lack of knowledge - I assume that he's not rheumatologist? Second, his lack of initiative in finding out what your options are, which after all, one should think is part of medical treatment (i.e., his job). I would recommend that you find another doctor so you can get treated for your pain. And if you don't have a rheumatologist, get one as soon as you can. You don't have to live in this kind of pain.