Is my prayer valid if I said in the first Tasleem, "Peace be upon the Prophet," then I realized my mistake and corrected it before the second Tasleem, then I said the second Tasleem and performed the prostration of forgetfulness?

Your prayer is valid, and what you did was correct, because you spoke inadvertently during the prayer, uttering only a few words, and you performed the prostration of forgetfulness (sujud (definition) al-sahw) for your oversight. This prostration is required for any addition or omission in the prayer. If you are a follower (ma'mum), you are not required to perform the prostration of forgetfulness unless you joined the prayer late (masbuq) and committed an oversight after you had separated from the imam to complete the remainder of your prayer.