It is unlikely that this cough is a serious underlying medical problem. However, the fact that you have traveled to the Dominican Republic puts you at higher risk of contracting tuberculosis, which is a chronic bacterial infection in the lungs that can be quite serious. Therefore, you should see your primary care doctor
. They will want to perform a PPD, which is a skin test for exposure to tuberculosis, to rule out that possibility. Assuming the PPD test comes back negative, then they will look for other causes of the cough.
As you have mentioned, sinus congestion or nasal allergies (allergic rhinitis) are high up on the list of what could be going on, especially if you have runny nose or nasal congestion or sinus pressure. Your doctor will be able to determine this for you, and will be able to prescribe medications to make the symptoms better. Common medications used for these symptoms include anti histamines like loratidine (Claritin) or cetirizine (Zyrtec) as well as nasal steroid sprays like fluticasone (Flonase).
There are other possibilities for the cough as well, including asthma
and gastro esophageal reflux (acid reflux
, heartburn). Your doctor will also be able to determine the likelihood that you have these conditions.