Even though you are paying via the CC annual fee you actually DO have a separate club membership. When I got mine the Club membership was set to expire ~3 months after the CC annual renewal. Was yours like that as well? And did you pay the renewal of the annual fee?
Assuming you did pay it recently it sounds like Chase didn't notify CO/UA of such so you didn't get the new card. I'd call Chase since it sounds like they need to push the data over to CO.