Middle English sore, from Old English sar "painful, grievous, aching, sad, wounding," influenced in meaning by Old Norse sarr "sore, wounded," from Proto-Germanic *sairaz "suffering, sick, ill" (source also of Old Frisian sar "painful," Middle Dutch seer, Dutch zeer "sore, ache," Old High German ser "painful," Gothic sair "pain, sorrow, travail"), which is perhaps from a PIE root *sai- "suffering" (source also of Old Irish saeth "pain, sickness"). Related: Sorely; soreness.
The slang meaning "angry, irritated" is recorded by 1738. Adverbial use, "painfully, so as to cause physical pain or injury" (as in sore afraid) is from Old English sare but has mostly died out (replaced by sorely), but adverbial use persists in the Modern German cognate sehr, the usual word for "very."