Is intoning the remembrance of God and repeating it with melodious chanting, accompanied by swaying, reverence, and weeping, in accordance with the Sharia, or is it better to repeat it without intonation?

There is no obligation to observe the rules of Tajwid in anything other than the Holy Quran, as there is no command to apply Tajwid to anything else. However, there is no harm in reciting supplications (du'as) and remembrances (adhkar) in a calm and moving voice that invites reverence (khushu') and reflection on their meanings, as long as it does not involve imitating the people of innovation (Ahl al-Bid'ah). It is recommended to adhere to the adhkar from the Quran and Sunnah, and what has been narrated from the righteous predecessors (Salaf as-Salih).