Yeah...if I said "see you anon," I'd get a response of "What?" or something about IT being Shakespearean...and of course it's not exclusively Shakespearean, but I think that's about as far back as most Americans read or know of in terms of English.
Hence why I have to stop myself punching the blackboard when explaining to clients that Shakespeare is not "Old English"...even though that's as "old" of a form of English as most Americans know, because Jamestown was founded in 1607, and OF COURSE, there was no meaningful history in terms of language, literature, or anything else before the first English settlement in Eventually-To-Be-America. ;)
That being said, I am now totally using "see you anon" for just that effect.
(And I'd again argue, as I have in other threads, a lot of this is due to exceedingly-poor education when it comes to how literature is taught and what literature is taught...literature, among other things, is supposed to help shape your worldview, and if your worldview extends no further back than Shakespeare, Jamestown, and the King James Bible, you're not only going to have a very Anglo-centric/Euro-centric view of history...I honestly think schools try too hard to get kids interested in very recent books while ignoring the classics--recent works can be great, but you can't appreciate them fully without the classics, and you're going to have a headache in college if you're an English major and DON'T know the classics...because sooner or later some professor's going to spring an author like my good buddy T.S. Eliot on you, and THEN you're going to be completely lost because you have no idea who Chaucer, Dante, or any of these other pre-Shakespeare authors are that he's referencing in "The Waste Land.")
But I digress.
As for other writers in Shakespeare's day--
"Methinks" isn't too special a word, I'd probably guess a lot of writers did use it as well...I can see Marlowe or Ben Jonson or Thomases Kyd, Dekker and Middleton (Thomas was apparently quite the name for playwrights in that era!) and so on all using it...no reason why they wouldn't, and it's a nice two-syllable word that would fit nicely into an iambic meter, meTHINKS, unstressed-stressed, so yeah, it probably saw some decent usage.
I can imagine it being used in Milton and the 1600s as well...and those sort of things don't go away overnight...
If I were to guess offhand I'd say that Austen's period--Late Enlightenment/Early Romantic--was around the time it started to become less commonplace in literature...I can't picture an Austen character really using that term, so either it became less widely used then or it was no longer a term that'd fit in "heightened speech" or literature dealing with the gentry and upper classes, since that's what Austen writes about (that and gossipy romance plots that end in marriage and simultaneously intrigue and irritate me--marriage does NOT equal happily ever after every single time, hell, Shakespeare and even Chaucer's Wife of Bath were more cynical and frankly realistic about that!--but I, again, digress.) :)