Hi Stephanie, I'm allergic to everything you are except dogs. We've recently bought a house with carpet, but luckily, there is hardwood floors underneath.
I've been getting allergy shots, which are covered by my insurance. You may be able to buy a Hepa air cleaner -- you can get one for your bedroom -- and there are also mattress and pillow covers that protect against dust mites. You'll have to check with your medical insurance to see if they cover this. The company I buy from is http://www.allergycontrol.com Your doctor is right. You do need to get rid of your carpet. That is where the most dust mites and animal dander is. Also, although I hate today this, air conditioning is much better for indoors than opening windows. It filters out many of the outside allergens.