I don't know much about Carnage, but is he known for being anything other than violent as fuck? Because the traits you used to describe Woody, are often the kinds of traits (wacko conspiracy theory BS), that you find in people who decide it's a good idea to shoot up massage parlors, and assault pizza shops to save the kids from the blood drinking democrats. And....I can EASILY see people like that, when given access to a super powered symbiote, it just fueling their mania into full blown death and...well...carnage. I mean they do that kind of shit IRL, so yeah, I can see someone like that doing just fine as a mindless killer.
Also, as others have pointed out, Woody became a household name, playing a homicidal maniac, so he's more than capable of doing it. Yes I know he was on Cheers, but I didn't really know him until Natural Born Killers, that's when he really became pop culture huge.