However, if a newspaper prints a story saying senator X said Y, and that claim was false, then the newspaper can be sued, regardless of where they got that information or which employee wrote the story.
Given your analogy, the question is, is Facebook more like a car manufacturer or a newspaper?
If a newspaper runs an advertisement that turns out to be false (say, a fake claim about a thing for sale), is it the newspaper or the advertiser who's at fault?
The advertiser. But if someone sues the advertiser and it turns out that the person listed by the newspaper as the advertiser isn't actually the advertiser, it's the newspaper's fault for not verifying their customers.
There are certain industries that have requirements for vetting customers, and if those requirements are not met then responsibility falls on the company and not the customer.
If a car manufacturer distributes a car that did not go through any inspections and it turns out the brake lines were hooked up incorrectly and consequently causes damage or hurts the customer then it's neither the customer or the dealership who are at fault, but the manufacturer for failing to properly inspect the product that they are making.
In the same sense, if a bar sells beer to someone without asking for their ID and they turn out to be underage, or a gun store sells a weapon to a convicted felon without performing a background check, liability falls on the company and not the customer.
I am not sure if this should apply to newspapers in respect to their advertising, but there are examples where this line of thinking does apply and this might be one of them.
Given your analogy, the question is, is Facebook more like a car manufacturer or a newspaper?