they'd start showing the first person eliminated's biggest baggage again. As in, "____, this is my biggest baggage." Now they just send them off and show a short clip of them making some lame wisecrack. I genuinely get curious of what that person's third case contains.
Is this something they recently started doing, or are those just old episodes?
Actually they do. In the earliest episodes this wasn't the format yet, and when the eliminated person went backstage they'd make the lame wisecrack you described. The producers of the shows agreed with you, and at some point in the first season they made the decision to show the "biggest bag."
The reason it seems like sometimes they do it and sometimes they don't, is because GSN has been airing the epsiodes out of order and mixing newer ones with older ones. Anytime you see the older format, take a look at the copyright at the end of the show - it will be 2010 and not 2011.