(Concern) I have a severe cough.

I caught a cold and was exhausted, with a runny nose and cough, and it was difficult. My immune system might have weakened, or maybe this cold is particularly severe lately. I think I have the worst cold I've ever had in my life. It should get better with time, but other symptoms have disappeared, yet the cough won't stop. The cough makes my throat sore, and my head hurts, and it's exhausting. I've taken medicine and drank ginger tea and tried my best, but the cough is too tiring. Should I go to the hospital?