Thanks everyone for the help and suggestions! I called our vet shortly after posting this and he thought it was post nasal drip. He recommended I give Cliff Benadryl. Cliff's symptoms actually got worse over the weekend... sneezing constantly during the day. gagging, and then he started getting yellow/greenish nasal discharge.
We took him to the vet yesterday. He concluded that he has a sinus infection which would be caused by some type of upper respiratory infection. He sent off a nasal swab to find out exactly what type of infection and the best type of antibiotics to treat it. That will take 5 days to get back.
In the meantime, he gave us 7 days worth of a broad spectrum antibiotic and told us to continue giving the Benadryl as well. After seeing the nasal discharge, I was very concerned about distemper even though he's been vaccinated. Our vet wasn't concerned though because he doesn't have a fever, has plenty of energy, is eating, stool is fine, etc. Phew!
He's on day 2 of the antibiotics and seems to be improving a bit. He still had quite a bit of nasal discharge this morning but the sneezing and coughing/gagging seems to be less frequent.
Thanks again for everyone's help!