Sorry to hear about this cough.
The temperature is fine.
The cough to persist this length of time is not.
Possible explanations include bronchitis (either due to an infection or allergic), heart disease
, collapsing trachea, lung foreign body, to name a few.
Definitely, even if it has reduced, the persistent cough is always an easy indication to warrant getting some simple chest x-rays just to be safe. The x-rays will also help to determine if a course of antibiotics are even necessary.
For allergic issues, it wouldn't hurt to give an antihistamine, but I would not rely on this treatment. X-rays would be the best.
Benadryl can be given at a dose of 1mg per pound of body weight. Keep in mind, Benadryl tablets and liquids come in different sizes. So, an approximately 25 pound dog can get a full 25mg tablet or a half of a 50mg tablet. I usually avoid the liquid Benadryl in larger dogs (you would have to give too much of it). You can give Benadryl every 8-12 hours.
Hope that info helps
I wish the best for both you and your pet. Please let me know if I can be of any further assistance.
I truly appreciate positive feedback if my advice has been helpful or informative.