It seems to me that Sauron went pretty full-out in LotR and there were no dragons involved in that. BUT. It says in The Hobbit that Smaug came from the North, which is - as far as canon goes post-Silmarillion - pretty much a desolate wasteland of nothingness, trolls and wights. There may be dragons there that were too far away for even Sauron to reach. Who knows?

Though now I have an image of Morgoth drawing in the dirt with a stick, being watched by Ancalagon.

Yeah, I don't really want them to bring up the One around Bilbo at all - it would make him think too much. But a flashback to Thrain, while disturbing, would be amazing. I think the dwarves do know. Perhaps not the bearers, but the people who witness them slowly going off the deep-end might have connected the two.