Fearful means terrifying, awful, and it also means afraid, full of fear and timorous. We saw a fearful animal is ambiguous, since you cant tell from the sentence whether the animal fears us or causes us to fear it. Making fearful even more elusive is its latest and increasingly frequent use to mean huge, great, usually applied to things thought unpleasant or bad, as in a fearful crash or a fearful cold.
Fearsome means dreadful, frightening, causing fear but also has an opposite sense, frightened, timorous, which causes confusion just as do the contradictory senses of fearful. Good advice: when you use either fearful or fearsome to mean frightened, state the cause of the fear, too (Shes fearful [fearsome] of the dark). Better still: dont use fearsome in that sense at all; use it for the horrible, not the timorous.
Fearfully, the adverb, has meanings similar to the adjective fearful, but it has gone on to develop into an intensifier as well: I have a fearfully bad cold. Fearsomely is an adverb meaning horribly, frighteningly, and it seems not to have developed beyond nonce word use as an intensifier.