www.alphadictionary.com

Thusly (adverb)

Alternative of thus.

This seems to be an Americanism, dating from the 1860s. It is hardly ever used in the UK. Apparently it was first used humorously, to imitate an uneducated speaker's attempt to sound correct.

Is this word used throughout the USA, or is it a regional thing? And does it still have an air of facetiousness?

I definitely hear thusly used as a synonym for thus in a serious manner. Can't offhand recall a specific, but I believe it mosly comes during an explanation. Perhaps it does have an advbial sense following the naming of a sequence of events, thusly showing how to accomplish something.