Is it permissible for me to pray the two rak'ahs of the Sunnah prayer of Dhuhr, then sit for the Tashahhud after one rak'ah, then say the Taslim, and then complete one rak'ah after that?

Your duty was not to make the salutations (taslim), but to stand up and perform another bowing (rak'ah) to make the rak'ah you performed even, and to perform the prostration of forgetfulness (sujud al-sahw). You must acquire the religious knowledge pertaining to your individual obligation (fard ayn), such as prayer and its conditions, and fasting.