When you do something despairingly, the action itself shows that you have no hope left. Someone may shake their head despairingly, look despairingly at a broken phone, or sigh despairingly — the -ly word rides along with the doing and tells you the feeling inside it. It is the how-word that matches the adjective despairing. Friends in a hurry are more likely to say “She looked so hopeless.” You meet despairingly in stories and in careful writing more than in chat. It is not a light “oh no,” and it is not a joke. Though actually, English already has despairing for the look or the mood itself (a despairing look). Despairingly tells you the manner, not the kind of thing. A despairing look is a kind of look; looking despairingly is looking in that hopeless way.