Is my prayer valid if I said the Taslim in the second Rak'ah of the Dhuhr prayer, then repeated the prayer for four Rak'ahs, and did not perform the prostration of forgetfulness (Sujud Al-Sahw)?

Your prayer is valid. If you remembered that you had given the salām before completing the prayer, you should have returned to your prayer with the intention, performed the last two rakʿahs, given the salām, and performed the prostration of forgetfulness (sujūd al-sahw). Rectifying a prayer that had an error is more preferable than starting a new prayer. Jurists have stated that drawing near to Allah with a patched and mended prayer is better than abandoning its mending or starting another prayer. Limiting oneself to it after mending is better than repeating it, because all good lies in following (the Sunnah (definition)) and evil in innovation.