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.
1 Milk may cover the back of the throat, but milk itself does not cause mucus.  In gastroesophageal reflux disease (GERD), gastric acid may back up from esophagus into the windpipe and throat. Burning stomach, chest (behind the breastbone), and throat, and coughing up white, thick, often frothy mucus are main symptoms.