Old English fullice "entirely; perfectly; completely;" see full (adj.) + -ly (2). Of similar formation are Dutch vollijk, German ööllig, Danish fuldelig.
Entries linking to fully
full adj.
Old English full "containing all that can be received; having eaten or drunk to repletion; filled; perfect, entire, utter," from Proto-Germanic *fullaz "full" (source also of Old Saxon full, Old Frisian ful, Dutch vol, Old High German fol, German voll, Old Norse fullr, Gothic fulls), from PIE root *pele- (1) "to fill." Related: Fuller; fullest.
The adverb is Old English ful "very, fully, entirely, completely" and was common in Middle English (full well, full many, etc.); sense of "quite, exactly, precisely" is from 1580s. Full moon, one with its whole disc illuminated, was Old English fulles monan; first record of full-blood in reference to racial purity is from 1812. Full house is 1710 in the theatrical sense, 1887 in the poker sense (three of a kind and a pair, earlier full-hand, 1850). Full-dress (adj.) "appropriate to a formal occasion" is from 1761, from the noun phrase.
-ly 2
common adverbial suffix, forming from adjectives adverbs signifying "in a manner denoted by" the adjective, Middle English, from Old English -lice, from Proto-Germanic *-liko- (cognates: Old Frisian -like, Old Saxon -liko, Dutch -lijk, Old High German -licho, German -lich, Old Norse -liga, Gothic -leiko); see -ly (1). Cognate with lich, and identical with like (adj.).
Weekley notes as "curious" that Germanic uses a word essentially meaning "body" for the adverbial formation, while Romanic uses one meaning "mind" (as in French constamment from Latin constanti mente). The modern English form emerged in late Middle English, probably from influence of Old Norse -liga.