Call your doctor
It is true that nose surgery is ALMOST always pain free; if your having so much pain you should call your surgeon and speak to him/her. Breathing through you mouth should not make a difference.
You are a brand new post-op and you should ask your surgeon about pain meds. It is normal to have some pain after surgery.
I would recommend calling your surgeon to discuss your pain control. There is a small possibility of a complication(s) that can lead to excessive pain. In the meantime, icing the nose can help.
Ask your surgeon!
You must trust your surgeon that may operated on you! None of us knows exactly what you had done or why you would be having pain. Only your doctor should advise you on taking narcotic pain medication.
Pain after rhinoplasty
Since you are very early in your post-op course, you should call your surgeon directly with these questions. They will be better suited to discuss your specific situation. In general, the pain will improve over the next few days and ice is one of the best things to reduce the swelling which causes much of the pain.
Best of luck