What is the ruling on the fast of one who intended to fast immediately before a nearby Adhan, knowing that another Adhan follows it? Is it obligatory to trust only the heard Adhan? And what is the ruling on prayer and fasting after the aforementioned Adhan?

Religious rulings, such as the obligation for a fasting person to abstain and the permissibility of prayer, are tied to the entry of the time, not to the Adhan (call to prayer), which merely announces it.

A Muslim woman must rely on reliable calendars or a trustworthy muezzin to accurately determine prayer and fasting times.

Your fast is valid if you made the intention before the Adhan of a muezzin you trust. Even if you doubt the entry of the time after eating, your fast is valid as long as you are not certain that you ate after the time had entered, as the default is the continuation of night.

As for prayer: If you pray a sufficient time after the Adhan of a trustworthy muezzin, such that you are certain the time has entered, then there is no problem. However, if you pray immediately after a questionable Adhan, such that the Takbirat al-Ihram (opening Takbir) occurs before the Adhan of other muezzins, then you must repeat the prayer, because you prayed before being certain of the time's entry. Whoever doubts the entry of the time should not pray until they are certain or it becomes highly probable to them.