Is it possible to print books/placards/newspapers/whatever in the Forgotten Realms? Or are all texts hand-written?
Yes, for at least a century or more
Printing presses are known to exist in the Realms for more than a century. There is a short story by David Cook named Patronage in the 1993 anthology Realms of Valor. The story takes place in Procampur in 1362DR and describes the surprise of a Tuigan lama, Koja, upon seeing a printing press for the first time and his attempt to get his A History of the Tuigan to be mass-published.
There is also a 4-part series of articles by Ed Greenwood, titled Small Presses of Waterdeep, published on the WotC website in 2003 (again the similar time period DR). These are still available on the web, where you can read about the printing presses and bookshops of Waterdeep.
Anyone can copy any book without legal penalty in Waterdeep, and printers amass libraries of chapbooks printed by their rivals so that they can plunder for ornaments and illustrations when a "new" book must be swiftly assembled.
Bookshops in Waterdeep tend to be crowded with dusty histories and volume after volume of adventure, romance, or bawdy sagas that are twenty to forty titles long.
While the Spellplague seems to have caused some harm, the printing business seems to have recovered quickly and the situation in 1490s is not much different in much of Faerun according to unofficial tweets by Ed Greenwood:
... all of the major port cities up and down the Sword Coast and around the Shining Sea, cities of wealth or rallying wealth like the cities of Cormyr and Sembia, and Westgate and Zhentil Keep, and centers of books and readership like Derlusk in the Border Kingdoms, had small, hand-operated printing presses that did more than just broadsheets ... by the 1420s DR.
Bookshops became fixtures of the Sword Coast port cities and all major Heartland trading cities and ports by 1475 DR, and places like Waterdeep, Silverymoon, Derlusk, Baldur’s Gate, and Suzail had local bestsellers and a marketplace of “here’s what’s coming” and “read a chapbook excerpt from the forthcoming new sequel to X by talented and famed Author Y” by 1478 DR.
In Waterdeep Dragon Heist (p. 181) we see the following text:
Lastly, no city in the world is as literate as Waterdeep. [...] The city has over thirty publishers of broadsheets in addition to chapbook printers and book publishers.
Broadsheets are, of course, the printed sheets that get folded down to produce the signatures of books and newspapers, etc. They can also be used, unfolded, as posters (which, historically, were what the news was printed on before the advent of newspapers).
For further edification, while technically not part of the Forgotten Realms, printing presses also exist in (at least some pocket dimensions of) the Shadowfell, suggesting that the technology is widely available in many of the various planes of existence.
From Curse of Strahd, in room W20 of the Wizard of Wines, (page 179):
In this chamber are a desk, a chair, a tall wooden cabinet, and a strange contraption that takes up most of the northern end of the room.
The contraption standing near the north wall is a printing press, which Davian Martikov uses to make wine bottle labels. The ink is made from wine and stored in bottles in the cabinet, along with pieces of parchment and jars of glue.