Hello,
From your symptom paronychia should be ruled out. Paronychia results from a breakdown of the protective barrier between the nail and the nail fold.
Treatment includes warm water soaks of the affected finger 3-4 times per day, oral antibiotics, topical antifungals and surgical intervention in more severe cases. My sincere advice would be to consult a dermatologist. Since it has not responded well to antifungals, so surgical intervention may be needed.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.