Is what the questioner did by exposing her cousin—who she discovered was sending the questioner's personal photos (the questioner's photos) as if they were the cousin's photos and engaging in sexual phone calls with young men—forbidden, and should she have concealed her?

If your cousin talks to foreign men about sexual matters and sends your pictures as if they were hers, this is a reprehensible and forbidden act. It opens the door to fitna (temptation/strife), and Islam has warned against taking "akhdan" (secret lovers/friends). If she was not openly committing her sin, then the obligation was to conceal her sin and advise her in the best way, and exposing her would be a sin on your part. However, if she was openly committing her sin, then there is no sin upon you for exposing her.