badly vs gravely

badly and gravely both are adverbs.

Word	Noun	Adjective	Verb	Adverb
badly	No	No	No	Yes
gravely	No	No	No	Yes

As adverbs, badly and gravely are synonyms defined as:

Other synonyms of badly include seriously, severely.

badly (adverb)	gravely (adverb)
(`ill' is often used as a combining form) in a poor or improper or unsatisfactory manner; not well	in a grave and sober manner
to a severe or serious degree	to a severe or serious degree
very much; strongly
with great intensity (`bad' is a nonstandard variant for `badly')
in a disobedient or naughty way
evilly or wickedly
unfavorably or with disapproval
with unusual distress or resentment or regret or emotional display
in a disadvantageous way; to someone's disadvantage
without skill or in a displeasing manner