I don't use Oxine - although I have some. So I might check into using it. I use a commercial egg wash & follow the manufacturer's directions for how to mix it. You shouldn't immerse the eggs in water or cleaning solutions. I wash them under running water, set on paper towels and pat dry before refrigerating.
They may keep longer without washing - but I wash anyway. All commercial eggs are washed - and we don't keep them around long enough for the whole bloom thing to be an issue. I want only cleaned eggs in the refrigerator with my other food.