Dearth is closer in meaning to lack and scarcity. These words can mean total or partial insufficiency. Usually an adjective is used with them to express total deficiency.
Ex – You display a complete lack of manners.
Absence is opposite of presence and thus it always means complete lack.
Ex – There is absence of bread in this store.
Shortage always means partial insufficiency, usually of an established amount.
Ex – There was shortage of medicine in the village.