Yea, sounds like kennel cough. Just make sure the new dog gets his shots, including the one for kennel cough. The dog with the bad cough most likely needs antibiotics and possibly a shot depending on the severity of the cough. A trip to the vet will do the trick. Sometimes, even after the shot and meds, the dog will continue to cough for about a month. My puppy seemed to get worse before he got better. It'll all work out.
Good luck!
Chrissy
Kennel Cough is highly infectious. It's possible he picked that up from another dog somewhere on his travels. I wouldn't think it's the food you give him causing his cough (?)