Is my prayer valid if I doubted the number of Rak'ahs (three or four) during the last Tashahhud of the Asr prayer, and I based it on the lesser number (three), then completed the Tashahhud and said the Taslim, and after that, I performed another Rak'ah and prostrated for forgetfulness (sujud al-sahw)?

You erred in what you did. The correct action would have been to perform another rak'ah (unit of prayer), then recite the tashahhud (testimony of faith), then perform two prostrations of forgetfulness (sujood al-sahw) before the taslim (salutation of peace), and then finally say the taslim. This is because whoever doubts the number of rak'ahs should base their prayer on the lowest certain number, and then perform the prostrations of forgetfulness. The proof for this is the noble Hadith: "Let him cast away doubt and build upon what he is certain of, then prostrate twice before he says the taslim." Since you said the taslim before performing the rak'ah, your prayer is invalid, and you must repeat it.