Only people who are admins of the Facebook Page on Facebook see the "Posted by Adam Smith" name (or whomever) on the Page Timeline. Non-admins of the Page do not see the "Posted by..." names. There is no cause for concern if the wrong admin name is listed on the post on Facebook, because only other Page admins see it.
The "Posted by..." name means that Social used that user's token to publish the post through Social onto Facebook. The token used does not necessarily match the post author name.
Let's say Jen Miller's authorization token expires tomorrow and another post is scheduled to go out on Friday. On Friday, Social will need to use another valid authorization token for that Facebook Page in Social, so Jen Miller will not be listed as the "Posted by..." admin name on Facebook.