In standard English, "cringe" is a verb: "It makes me cringe." Or it can be a noun: "My smile turned into a cringe."
"Cringey" is an informal English adjective meaning "cringeworthy".
Current slang among young people has "cringe" being used as an adjective as in your example, correctly written as: "It's cringe seeing people's overreaction."
So, it's fine to use "cringe" as an adjective as long as you understand that it's slang, not something to be used in formal contexts.