Does the barometric pressure cause headaches or is it just sinus pressure due to allergies?

Yes and yes. Weather changes can trigger many types of headaches. There is a specific type of sinus headache, called barosinusitis, that can be triggered by barometric pressure changes.
Absolutely! Barometric pressure changes is a common trigger for nasal symptoms, or vasomotor rhinitis. Other triggers for vasomotor rhinitis include sudden temperature changes, irritants like air fresheners/smoke, exercise, or cold air. Since traditional allergy pills do not work for vasomotor rhinitis, it is best to see a physician to get a prescription nasal spray for this type of trigger.