Limberly does not come across as being exotic or arcane; it seems like an everyday word. But if your experience echos mine, we rarely hear limberly in our daily lives--much less in our every day lives. I agree with J. N. Hook that for "some reason limberly appears much less often than its companion, the adjective limber, yet it should be equally useful: 'Although he's in his fifties, he plays hardball vigorously and limberly'." 
Webster's Third Unabridged Dictionary is one of the rare lexicons to record that the adverb limberly exists and is worthy of definition. I will grant you that most readers of W3 do not really need a definition for limberly, for we intuit that such an adverb probably exists and probably means "in a limber manner." It's just inexplicably odd that such a comely, useful word is so infrequently invited into our Big Party of Words.
"I was impressed at how limberly and politely David Gregory responded to Tony Snow's accusation that Gregory had broken White House Press Room protocol by asking a question with an apparent partisan twist." --B'n' J'
What limberly needs:
Publicity! Maybe if there were a hit television sit-com about a circus acrobat named "Limberly Betty". . . .
 Hook, J. N. The Grand Pandrum. New York: McMillan, 1991, 250.