Heedlessly is how you do something when you are not paying attention to a warning, a rule, or a danger. Heedless describes a person or an attitude; heedlessly describes the action: she drove heedlessly, they spent the money heedlessly. It sounds a little bookish. In everyday talk people more often say without thinking, without looking, or without paying attention. Though actually, the old word is useful when you want to show that the person had a chance to notice — and still went ahead.