Blepharitis is an inflammation of the eyelids. It can be due to infection, allergy or irritation. In your case that house dust may have some influence on your condition. Burning in the white portion of both eyes might indicate that conjunctiva is also affected (blepharo-conjunctivitis). Treatment includes local antibiotics and anti-inflammatory drugs (steroids). Blepharitis can’t cause blindness except if infection spreads toward deeper layers of the eye.
You can request microbiological examination of a sample taken from the conjunctiva in order to take a proper antibiotic if some bacteria is detected. You probably should continue with the steroid eye-drops. You may also want to take some painkillers for the eye-pain.