If you know an author's real name and it differs from the handle (the author's account name), its best to denote the handle in square brackets following the authors name.
Twitter post
In-text:
(@jkbjournalist)
Work Cited list:
Brown, Julie K [@jkbjournalist]. "NEW: Miami Herald finds Covid cases surging since Florida has reopened, state downplaying threat." Twitter, 12 June 2020, 3:58 p.m., www.twitter.com/jkbjournalist/status/1271444548024176641.
Facebook post
In-text:
(BBC News)
Works Cited list:
BBC News. “The 17.7% increase in May added to hopes that the country's economic recovery could be faster than predicted.” Facebook, 12 June 2020, 10:00 a.m., www.facebook.com/bbcnews/.Accessed 26 June 2020.