I received an Aqeeq ring as a gift from one of my friends. However, since I do have multiple other rings, I am not using this one. Can you please tell me if it is permissible to put the ring in flowing water just like we do with other muqadas auraq? Or is there any other way to properly dispose it off?

You can gift to other In’Sha’Allah you both will get sawab for this act.