Was what I did correct, in that I made a mistake in uttering the Taslim in the Dhuhr prayer, so I said "Peace be upon you, O Most Merciful of the merciful," then I corrected myself and said "Peace be upon you and the mercy of Allah," then I performed the prostration of forgetfulness after the Taslim, and felt a release of wind, and when performing ablution for the Asr prayer, I performed the two prostrations of forgetfulness for the Dhuhr prayer?

As long as you caught yourself and performed the taslim (salutation of peace) correctly, your prayer is complete. It is prescribed for you to perform the prostration of forgetfulness (sujud al-sahw), but it is not obligatory. If you felt something (like an impurity) during prostration but it did not reach the level of certainty, then you are still in a state of purity, and your prostration is valid. However, if you became certain that wind was emitted and your prostration was after the taslim, then your prayer is valid, and nothing is required of you. Nevertheless, it is sunnah to perform wudu (ablution) and then perform the prostration. Performing the prostration during another prayer's time is not beneficial, but the prayer is not invalidated by its omission.