According to this guide, in case you sell a ring you can pick up another one in the same spot where you found the one you sold in your next playthrough.
Make sure you don't sell a ring you got as a covenant reward, since your covenant rank carries over to NG+ and you probably won't be able to get the covenant reward a second time.
In case you ever sell a ring you can't get again, you can still unlock the achievement by trading the required ring with another player. Rings can be picked up by other players and will count towards the achievement.
Here is a page on reddit where you can request trades with other players in case you don't know anyone with the required rings.