Sounds like kennel cough to me. My family has adopted 3 dogs from different shelters in the last couple months (separately), and all three have had kennel cough; one died from it (advanced to a severe case of pneumonia, we only had her for 4 days

).
Luckily, our other two rescues were diagnosed early enough and have been successfully treated. It's an easy fix if you catch it early. Take your pup to the vet!