From what you write, I'm thinking that you have allergies that involve cats, but are not limited to cats. It would make sense to call your allergist's office and ask about the carpeting and anything else that came with the house (drapes, etc.). Pick their brains and get lots of suggestions. If you end up replacing things and it helps your health, it'll be money well spent, but make sure it will be worthwhile to do first.
I have some friends who had to take out all the (relatively new) carpeting at their house simply because there had been animals in the house previously. They went with hardwood floors instead.
Since I'm a cat-lover, I'd hate to see you turn out your four-footed family member unless it was absolutely the guilty party (and then you DO have to find it another home), and from your post I'm not sure of that. So I'm rooting for the carpeting to be the bad guy. (Non-animal lovers just don't understand this sort of thinking, so don't expect them to.)
By the way, I found some online articles about cat dander. One of them is:
http://cats.lovetoknow.com/Removing_Cat_Dander
Hope this helps.