Origins of "Seedy"

seed·y ˈsēdē adjective 1.sordid and disreputable.
"his seedy affair with a soft-porn starlet" synonyms: sordid, disreputable, seamy, sleazy, squalid, unwholesome, unsavory

shabby and squalid. "an increasingly seedy and dilapidated property" synonyms: dilapidated, tumbledown, ramshackle, falling to pieces, decrepit, gone to rack and ruin, run-down, down-at-heel, down-at-the-heel(s), shabby, dingy, slummy, insalubrious, squalid;
antonyms: classy

My wife and I live in NYC and we like to joke that this terms comes from Avenue C and Avenue D, a particularly run-down and, well, seedy part of Manhattan. I doubt this is true. What is the origin of the word and how did it come to mean "sordid and disreputable" and "shabby and squalid"?

Nothing so imaginative, I’m afraid. The Oxford English Dictionary says it is ‘apparently in allusion to the appearance of a flowering plant that has run to seed.’

"He was soiled and seedy and fragrant with gin."- Mark Twain

I had hoped to uncover something more... such a rich word.

First Known Use of SEEDY 1574

It's a mystery how the word seedy came to mean darkly rundown, slummy, and seamy, but it probably came from the appearance of flowers after they've shed their seeds. That's when they start to lose their color and eventually die.

seedy (adj.) mid-15c., "fruitful, abundant," from seed (n.) + -y (2). From 1570s as "abounding in seeds." Meaning "shabby" is from 1739, probably in reference to the appearance of a flowering plant that has run to seed. Related: Seediness.

seedy: 1440, "fruitful, abundant," from seed (n.). Meaning "shabby" is attested from 1739, probably in allusion to the appearance of a flowering plant that has run to seed.

You must understand I have been very seedy indeed, quite a dead body; and unless the voyage does miracles, I shall have to draw it dam fine. Robert Louis Stevenson

He tramped the streets of Brussels, at first in seedy clothes and at last in filth and horrible rags. - The Pools of Silence, H. de Vere Stacpoole

And half with weakness. "Sir," I said,
— But with a mien of dignity
The seedy stranger raised his head:
"My friends, I'm Santa Claus," said he.
"The Hard Times In Elfland [A Story of Christmas Eve]" by Sidney Lanier

Yesterday I felt rather seedy, having a touch of Cuban fever, my only unpleasant reminiscence of the Santiago campaign. - Roosevelt, Theodore. 1919. Theodore Roosevelt's Letters to His Children: Quentin's First Fall.