A small correction there. The 'Eustachian tubes balance the pressure between the (inner) ear and the airway', not the sinuses. They have no direct connection to the sinuses. Clearing a blocked sinus is extremely difficult, and can
sometimes be achieved by the Valsalva action
Valsalva maneuver - Wikipedia, the free encyclopedia but not reliably. Pain from descent with a completely blocked sinus can be excruciating. I once suffered this from a circuit detail in the RAF when descent from 1000' to ground level caused intense pain.
Do not treat a bad head cold lightly in aviation. fujii is spot on.