Probably you'd notice a bit of extra time posting for the signature to be added, but that's about it, the responsibility for verifying the signature would fall to the owners of the social media site and in the circumstances where someone asks for a verification, basically imagine it as a libel case on fast forward, you file a claim saying "I never said that", they check signatures, they shrug and press the delete button and erase the post, crossposts, and if it's really good screencap posts and those crossposts of the thing you did not say but is still being attributed falsely to your account or person.

It basically gives absolute control of a person's own image and voice to themself, unless a piece of media is provable to have been made with that person's consent, or by that person themself, it can be wiped from the internet no trouble.

Where it comes to second party posters, news agencies and such, it'd be more complicated but more or less the same, with the added step that a news agency may be required to provide some supporting evidence that what they said is not some kind of misrepresentation or such as the offended party filing the takedown might be trying to insist for the sake of their public image.

Of course there could still be a YouTube "Stats for Nerds"-esque addin to the options tab on a given post that allows you to sign-check it against the account it's attributing something to, and a verified account system could be developed that adds a layer of signing that specifically identifies a published account, like say for prominent news reporters/politicians/cultural leaders/celebrities, that get into their own feed so you can look at them or not depending on how ya be feelin' that particular scroll session.