You have identified the two most likely possibilities causing your symptoms.
Firstly, dry coughs are typically viral coughs and may persist for some time.
Alternatively, you might have developed a form of allergic rhinitis (allergies to the airways) giving rise to the cough in a similar way that mild asthma might.
If the problem is a viral infection, I would suggest you give Echinacea a trial, if you have not already done so.
If specific treatment is required for your dry cough, try pine cough syrup.
For allergic rhinitis, I would suggest using Pollinosan, a remedy for hayfever as well as allergic rhinitis.
You might also benefit from staying away from cow’s milk and other dairy produce for the time being, and of course, avoid being in smoky environments.
Remember that if your symptoms persist, you should make an appointment to see your doctor.