Drink more fluids. Water helps loosen mucus and soothe an irritated throat. Dry, hacking coughs respond to honey in hot water, tea, or lemon juice. Do not give honey to children younger than 1 year of age. Elevate your head with extra pillows at night to ease a dry cough. Try a cough drop to soothe an irritated throat.
That is actually true to a certain extent. Most of the time you need to have some sort of sensitivity/allergy to dairy. But dairy products can increase the thickness of your phlegm, therefore can clog up your sinuses, and eventually cause the sinus pressure.
The sinuses drain through channels in the nose. When a sinus becomes inflamed, usually as the result of an allergic reaction, a tumor, or an infection, the inflammation causes swelling and increased mucus production and the channels can become blocked. The increased pressure in the sinuses causes a pain similar to that of a headache.