A Toy Story 3 fan theory speculates Woody would've essentially become Lotso, had the events of the original Toy Story played out differently. For all his positive qualities, Woody has a lot of insecurities stemming from his relationship with his first owner, Andy, many of which have served as the narrative driving forces behind the Toy Story movies. Whether he's jealous of new arrival Buzz Lightyear, wrestling with his own impermanence, struggling to embrace change, or trying to find a purpose for his continued existence, Woody's emotional complexity is a big part of what makes him such an endearing and relatable character.
With the exception of Sid Phillips from the first film, the antagonists in the Toy Story series have take on the forms of toys that are either struggling in ways that parallel Woody's issues at any given time or have turned to darkness (in some sense) and act as cautionary tales for the pull-string cowboy doll, making them effective foils to him. That's certainly the case with Toy Story 3's Lots-O'-Huggin' Bear aka. Lotso, an elderly, strawberry-scented, pink teddy bear and the leader of the toys at Sunnyside Daycare Center.
While he's initially gentle and caring towards Woody and the rest of Andy's toys after they're accidentally donated to Sunnyside, Lotso eventually reveals himself to be a cruel dictator who rules the daycare center with an iron (but furry) fist. As Woody later learns, Lotso lost his way after being separated from his owner, Daisy, and getting back to her, only to learn he'd been replaced with a brand-new Lots-O'-Huggin' Bear. Per the fan theory posted by a user on Reddit, "Woody would have turned out just like Lotso'", had Andy (for one reason or another) replaced him with another Woody doll in the first Toy Story.
There are some striking similarities between Woody and Lotso's backstories; they both went through difficult journeys to make it back to their original owners at some point prior to the events of Toy Story 3, with the difference being Woody got a happy ending and matured along the way, whereas Lotso did not and was forever traumatized by what happened. This isn't an accident, either, and the creatives behind the film clearly designed Lotso to serve as Woody's dark reflection (as they basically confirmed on the film's official commentary track). The first Toy Story movie even offered a glimpse of what a "dark Woody" would look like in the scene where Woody tries to knock Buzz behind a dresser in Andy's room, in the hope he'll be stuck and forgotten there. Obviously, that's not the same thing as acting like a prison warden the way Lotso does, but it does suggest there's a real possibility Woody would've gone in a similar direction, had things not worked out for him.
Interestingly, on the movie's DVD commentary, Toy Story 3 director Lee Unkrich revealed test audiences had wanted Lotso to redeem himself by turning off the incinerator and saving Woody and the other toys during the film's climax, but he and his collaborators decided it wouldn't really work. This highlights another major difference between Woody and Lotso: whereas Woody, time and time again, has proven capable of recognizing his mistakes and is willing to make hard choices to fix them while helping others, Lotso (as one of Daisy's other toys, Chuckles, glumly tells Woody) only cared for himself after what happened with Daisy and refused to change, even when Woody confronted him with the truth. In that sense alone, the Toy Story 3 villain really is "dark Woody".