Does the [practice of] saying "Allahu Akbar" (Takbir) stop between the Adhan and Iqamah, and before Fajr during the time of seeking forgiveness?

The absolute takbir (proclamation of Allah's greatness) on the two Eids is a recommended Sunnah. Stopping it is permissible because it is not obligatory. As for what is most virtuous, it varies depending on the questioner's condition, their emotional state, and their heart's disposition. This falls under the jurisprudence of virtues, not legal rulings. A believer is keen to combine virtues and chooses what suits their time and their own state, as the Prophet (peace and blessings be upon him) said: "Undertake such deeds as you are capable of, for Allah does not grow weary until you do."