Azan is recommended while Tashahhud is obligatory. Tashahhud is part and parcel of Namaz and if Tashahhud is wrongly said, Namaz will be void and invalid.
Tashahhud was taught to us by the Proohet Muhammad (SAWA) and Ahlul Bayt (AS) with ordering us to (Pray as you saw me praying) صلّوا كما رأيتموني أصلّي
Imam Jafar Al-Sadiq (AS) explained to one of his students; Hammad Ibn Easa all the steps of Namaz including Tashahhud ( ASHHADU ANLA ELAAHA ILLALLAH WAHDAHU LAA SHAREEKA LAH WA ASHHADU ANNA MUHAMMADAN ABDUHU WA RASOOLUH ALLAHUMMA SALLI 'ALA MUHAMMAD WA AALI MUHAMMAD as we read it in al-Kaafi, 3/311.
We are not allowed to add to Tashahhud but we are allowed to pray any prayers as supplication after Tashahhud including the names of the Infallible Imams like ALLAHUMMA SALLI 'ALA AMEERIL MO'MINEEN.