does dairy clog sinuses
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.
Milk and other dairy products are often blamed for additional production of mucus that can cause sinus problems but this may by a myth. A study published in The American Review of Respiratory Disease did not find an association between milk consumption and increased nasal secretions.