This is very frustrating for us too. Ever since this new interface was rolled out, it's been difficult getting used to it. I spent several minutes one day on the mobile app just trying to find Messages in the menus. It was a small envelope icon on the top right! Weird. Anyway, we sent a message to a Seller, and we cannot find a Sent folder to reference this message. There's NO way we would go back to Classic, because we insist on all messages being in the SMS format for easy conversation.
Update: it looks like the reason we didn't see a message thread in the new interface is because that Seller we sent a message to ridiculously does NOT accept eBay messages! We didn't know that was the reason at first though, because eBay merely presented a pop-up error stating "something went wrong, try again" (which doesn't imply a seller refuses messages... it seems more like a system glitch). Anyway, as long as the seller accepts messages, the SMS style messaging thread should show up in the new interface.
