The prescriptions for Clavamox and Temaril-P were a decent first guess at treatment for Teddy, but unfortunately there are a lot of other types of diseases that are likely to cause a 12-year-old dog to cough that won't get better on either of these medications.
In fact, the only diseases that would respond to that kind of treatment would be a mild bacterial infection (not common), or an allergy (also not a common presentation for allergy).
Some other diseases that could cause coughing in a dog like Teddy that are much more common and would require different treatment include:
and/or heart failure
-bacterial pneumonia (non-responsive to Clavamox)
I'm sorry to be so alarming, but I'm worried about Teddy. As you can see, nearly all of the diseases on the list are very serious.
So, yes please, if you are able, please take Teddy to the emergency vet. I'm eager for him to receive care right away, and he deserves an actual diagnosis instead of just a guess at what the problem might be.