My RA has been in remission since a few months into the gluten free diet. For complete relief I found I also had to drop soy but that may not be the case for you. I was also prescribed the same drugs you were (and loads more!} and found them not to help at all. I hope you get some relief from the diet but it can take time.
If you need a diagnosis of celiac to keep you strict on the diet do keep on the gluten until after you are done testing.