Hello, I'm Dr. Kara. I have over 20 years of experience as a veterinarian and I'd like to help with your concerns about Tonka's cough, nasal stuffiness and wheezing.
I suspect his cough started because of irritation from something causing increased mucous production and dripping down the back of his throat.
Dogs very rarely get nasal stuffiness from allergies, they usually get itchy skin and ears when they are allergic to something, but we cannot completely discount that as a possibility.
More commonly we see nasal inflammation secondary to dental disease (as some of the tooth roots end up near the nasal cavity). If he has a bad tooth, periodontal disease or gum disease this is possible.
Other possibilities are accidentally inhaling foreign bodies such as grass awns or even grass blades or pine needles, a nasal polyp or a fungal infection. A tumor at his age would be very unlikely, but possible.
At this point because he isn't responding to the medications given I recommend diagnostics to look further into his stuffiness.
The next steps would be a nasal scope to look for a mass or a foreign body, radiographs of his tooth roots to look for signs of an infected root and radiographs of nose and sinuses to look for signs of an infection or mass and possibly cultures (bacterial or fungal) if signs of infection are present.
At home you can run a humidifier or bring him into the bathroom with you when you run a hot shower or bath. The steam thins the mucous and soothes his airways so he can breathe more easily.
Best of luck with your fellow, please let me know if you have any further questions.