Ask the experts
We have no doubt that you had a sore throat. We have no doubt that you are still coughing. But if you have been taking an antibiotic for a strep throat, it may not be that you have (or have ever had) a strep throat, a true strep throat, one that due to a strep bacteria.
A true strep throat that is treated with an appropriate antibiotic (one to which it is sensitive) should, as a rule, be over and done with within a day (24 hours).
The fact that your sore throat did not clear up promptly goes very much against it being a strep. And the fact that you are still coughing 5 days later goes even more strongly against it being a strep.
Sore throats can be due to many causes including, very prominently, a hoard of viruses. A viral sore throat will not respond to an antibiotic and can cause inflammation of the airways (bronchitis). Therefore, a virus that causes a sore throat is quite capable of causing a cough.
Unless there is clear-cut, definite, convincing laboratory proof of a strep, our vote goes to a virus as the best bet.
If your symptoms persist, you should contact your doctor. Hope you feel better.