I don't know if this goes for specific games or if it goes for the whole site, but how do you apply to be a verifier in a particular game? Do you have to do it in the game's discord server or do you just have t ask in the forums of the specified game?

Usually, you can find applications to fill out in the game's discord or forum. Keep in mind that applications need to be open in order to apply.

generally just ask existing mods in DMs,

most games that aren't extremely well ran are always happy to have more verifiers. That being said, you actually need to run the game to be a verifier, don't just go asking random people.