Sauron possessed the 9 rings, as some of the 7, as you correctly recall.
Unfinished Tales:
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...mightiest servants, the Ring-wraiths, who had no will but his own, being each utterly subservient to the ring that had enslaved him, which Sauron held
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...were entirely enslaved to their Nine Rings, which [Sauron] now himself held
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And in LOTR:
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...the Nine [Sauron] has gathered to himself; the Seven also, or else they are destroyed
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You saw the Eye of him that holds the Seven and the Nine
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My thoughts as to why he had them, is what Tolkien wrote in
Letter 246 (another place where I believe Tolkien mentions Sauron held the 9 rings). And that is Sauron had a way to control his Nazgul, even without the One Ring. In Letter 246, Tolkien sets up a hypothetical scenario about Frodo the "ring-lord" claiming the One Ring, in the Sammath Naur, but the Nazgul would still take Frodo from Mount Doom and bring him to Sauron. As it was their 9 rings that that Nazgul were enthralled to, not the One.