Sam Kelly, at Literary Hub, asks “Did Shakespeare Write Hamlet While He Was Stoned?”:

That’s right, Shakespeare was a stoner. I’m not making this up—they found the evidence in his backyard. Back in 2001, some anthropologists got permission from a museum to borrow twenty-four clay pipe fragments that had been dug up in the small town of Stratford-upon-Avon, where Shakespeare used to live. Using state-of-the-art forensic technology, the anthropologists discovered cannabis residue on eight of them—including several from Shakespeare’s backyard garden—that dated back to the late 1500s/early 1600s, around the time he actually lived there.

Kelly admits the evidence is circumstantial:

Pipes with cannabis residue were dug up in Shakespeare’s garden, but that doesn’t necessarily mean Shakespeare is the one who put them there. (Hey, maybe they were planted in his garden by Sir Francis Bacon, right?)

Kelly also suggests that one of Shakespeare’s sonnets “comes pretty close” to “admitting that he smoked cannabis”:

Professor Thackeray, the same anthropologist who performed the pipe analysis, points to Shakespeare’s Sonnet 76, which talks in part about the act of writing […]

The wording in Sonnet 76 is pretty convincing. Many scholars believe the “noted weed” in which the author finds “invention” is a reference to cannabis and its ability to stimulate creativity.

I’m not sure who those “many scholars” are, but “weed” in Sonnet 76 has long been understood to mean “clothing or style of dress” (Oxford English Dictionary: “an item of clothing, a garment”). This usage is even seen in Two Gentlemen of Verona, where Julia disguises herself as a man and asks her waiting-woman Lucetta to “fit me with such weeds / As may beseem some well-reputed page.” It is not a reference to marijuana; the earliest use of weed in that sense is from 1898, almost three hundred years after Shakespeare wrote this sonnet.

“Noted weed” means “recognizable or familiar clothing or style of dress,” or, in the context of the sonnet, “familiar words and a recognizable writing style.”

This theory of “noted weed” being a reefer reference has apparently been floating around a long time. Hilary Hanson wrote about it for Huffington Post in 2015, in which she spoke to James Shapiro:

James Shapiro, a Columbia University professor who has published multiple books about Shakespeare’s life isn’t so convinced that Shakespeare was a stoner. […]

He’s especially skeptical of Thackeray using Sonnet 76 as supporting evidence, which Shapiro called a “really lame interpretation” of the poem.

“The line ‘keep invention in a noted weed’ is referring to weeds as dressing up, as clothes,” Shapiro said. “The poem is about dressing up language in a certain way and you really have to be insensitive to the poem to force the reading [to be about marijuana use].”

Back to Kelly at LitHub:

Some also think the phrase “every word doth almost tell my name” is a sly reference to the fact that “shake” (as in Shakespeare) is another word for cannabis—specifically, the scraps left over after cannabis buds have been plucked and packaged.

This is such a stretch that it defies logic and requires no rebuttal. You might as well argue that “tell my name” is a “sly reference” to Destiny’s Child. And the less said about Thackeray’s interpretation of “compounds strange” the better. (“Thackeray believes [Shakespeare] chose to avoid using other, more dangerous drugs, such as cocaine, because they were ‘compounds strange’ that could prove harmful.” I mean, c’mon. It’s clear from context that “compounds strange” is about composing unique word combinations.)

So, no, Shakespeare wasn’t seeking the sticky icky in Sonnet 76. He’s explaining why he hasn’t changed his writing style, which is recognizably his.