All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
It is better for you to say the athaan in all cases. Maalik ibn Al-Huwayrith, may Allah be pleased with him, said, "The Prophet, sallallahu ‘alayhi wa sallam, said to us, 'When the time for prayer comes, one of you should proclaim the athaan for you..." [Al-Bukhari, Muslim, Ahmad, Abu Daawood, An-Nasaa'i, At-Tirmithi and Ibn Maajah]
As to whether you are obliged to say the athaan or not, it is not an obligation, especially if you are in a city or village in which the athaan is proclaimed in its mosques. You are not obliged to say the athaan even based on the scholarly view that proclaiming the athaan is a collective obligation. This is because the athaan is already proclaimed in the mosques and, accordingly, the collective obligation is fulfilled. The majority of the scholars held that saying the athaan is a recommended act of Sunnah (and not an obligation).
Imaam An-Nawawi said:
"Views of scholars on the athaan and the Iqaamah (second, small call to prayer): the famous view of our school of Fiqh is that they are recommended for all prayers in residence and travel and for the individual and congregation. It is not an obligation to proclaim the athaan and Iqaamah in any given situation; if they are left out, the prayer is valid regardless of whether that is done by an individual or a congregation. This is the view held by Abu Haneefah, his followers (the Hanafis), and Is-haaq ibn Raahawayh. As-Sarakhsi cited it from the majority of scholars. Abu Al-Munthir held that they are obligatory for the congregational prayer in residence and travel. Maalik held that they are obligatory in the mosque in which the congregational prayers are held..." [Al-Majmoo‘]
We would like to highlight two points in this regard:
Firstly, the Muslim man is enjoined to perform the prayer in congregation in the mosque if the mosque is close enough for him to hear the athaan without loudspeakers. According to a group of scholars, it is an obligation on him to perform the prayer in the mosque in this case. He is sinful if he misses the prayer in the mosque without a valid reason. Please, refer to fatwas 82538 and 294724.
Secondly, it is recommended to perform the prayers at the beginning of their time because it means hastening to obey Allah. If you are used to delaying the Thuhr prayer near the end of its time, as understood from the question, then we advise you to pray it at the beginning of its time, following the Sunnah of the Prophet, sallallahu ‘alayhi wa sallam. Jaabir ibn Samurah, may Allah be pleased with him, said, "The Prophet, sallallahu ‘alayhi wa sallam, used to pray the Thuhr when the sun had passed its zenith." [Muslim]
So, perform the Thuhr prayer at the beginning of its time unless you are attending the congregational prayer at the mosque and it is a very hot day. In this case, it is recommended to delay the Thuhr until the heat becomes less intense. This is what the scholars call "Ibraad" or seeking coolness, based on the hadeeth in which the Prophet, sallallahu ‘alayhi wa sallam, said, "If it is very hot, then pray when the weather is cooler, as intense heat is a gust from Hell." [Al-Bukhari and Muslim]
If you perform the Thuhr at home instead of at the mosque, then you should perform it at the beginning of its time even on very hot days because there is no hardship if you are not going to the mosque. Al-Haafith said:
"The majority of the scholars said that it is recommended to delay the Thuhr prayer in the case of extreme heat until the weather is cooler and the intensity of the heat is broken. Some scholars specified that this applies to the congregational prayers only and that it is better for the person performing the Thuhr individually to hasten to perform it at the beginning of its time. This is the view of most of the Maalikis. Ash-Shaafi‘i held the same view but restricted it to hot countries. He underlined that the congregation in this context refers to the people who perform the prayer in congregation at the (distant) mosque. However, if the mosque is nearby or the road to the distant mosque is shaded, then it is better for them to perform the prayer at the beginning of its time. The famous view of Ahmad is that it is recommended to delay the Thuhr prayer in both cases, with no specification or restriction. This is also the view of Is-haaq, the scholars of Kufa, and Ibn Al-Munthir."
Allah knows best.