Praise be to Allah.
The Sunnah is to hasten to break the fast , which means breaking fast straight after the sun has set. Delaying breaking the fast until after the stars have come out is the practice of the Jews, and the Rafidis (Shi’ah) followed them in that.
So it is not correct to delay breaking the fast deliberately until later in the evening or to delay it until the end of the adhan. All of that is contrary to the teachings of the Prophet (peace and blessings of Allah be upon him).
Sahl ibn Sa’d narrated that the Messenger of Allah (peace and blessings of Allah be upon him) said: “The people will continue to be fine so long as they hasten to break their fast.” (Narrated by al-Bukhari, 1856; Muslim, 1098)
“This hadith means that we are encouraged to hasten to break the fast after ascertaining that the sun has indeed set. What this means is that the ummah’s affairs will continue to be in order and they will be fine so long as they adhere to this Sunnah. But if they delay breaking the fast, that will be an indication of some misguidance that they are following.” (Sharh Muslim, 7/208)
Abu Awfa (may Allah be pleased with him) said: I was with the Prophet (peace and blessings of Allah be upon him) on a journey, and he fasted until the afternoon. Then he said to a man, “Go and prepare a specific kind of food for me.” The man said, “Why not wait until it is evening?” He said, “Go and prepare (that food) for me, and when you see that night has come from this direction, then the fasting person should break his fast.” (Narrated by al-Bukhari, 1857; Muslim, 1101)
Abu ‘Atiyyah said: Masruq and I entered upon ‘Aishah and said: O Mother of the Believers, there are two men among the Companions of Muhammad (peace and blessings of Allah be upon him), one of whom hastens to break his fast and pray, and the other delays breaking the fast and prayer. She said: Who is it that hastens to break the fast and to pray? We said, ‘Abd-Allah (meaning Ibn Mas’ud). She said: That is what the Messenger of Allah (peace and blessings of Allah be upon him) used to do. (Narrated by Muslim, 1099)
Al-Hafiz ibn Hajar said:
“One of the reprehensible innovations that have appeared in our times is giving the second adhan approximately twenty minutes before Fajr in Ramadan , and extinguishing the lamps which indicate that it is haram to eat and drink for those who want to fast. Those who do this claim that they are erring on the side of caution with regard to an act of worship, but no one knows of this practice except a few. As a result of that, they have started not to give the adhan until a while after sunset, so as to be certain of the time, as they claim. So they delay breaking the fast and they hasten to eat suhur, thus going against the Sunnah. Hence they do little good and do a great deal of evil. And Allah is the One Whose help we seek.” (Fath al-Bari, 4/199)
And Allah knows best.