That could just mean that he completely controlled them, not had them in his possession. i.e. Held in his power, not his hand.
I always assumed that they still wore them but you may have a point there Muffer. And true, there was no mention of a dropped ring after the Witch King dies. On the other hand there was no mention of a lot of things...
As for the dragons: they may have consumed the rings and later excreted them. There's a joke in there somewhere about passing rings on but I can't seem to find it. And those dragons could have been killed too. All speculation of course.
I'm sure that Barahir was a common name in those parts/times. It's possible that there was more than one "Ring of Barahir". Nine? Maybe it was
really common. Or Sauron had no imagination when it came to naming things.
