Praise be to Allah.
The beginning of the month of Ramadan can only be proven if the new moon is sighted, or the month of Sha‘baan is completed with thirty days, because of the report narrated by al-Bukhaari (1909) and Muslim (1081) from Abu Hurayrah (may Allah be pleased with him) who said: The Prophet (blessings and peace of Allah be upon him) said: “Fast when you see it and break the fast when you see it, and if it is cloudy then complete the number of days of Sha‘baan as thirty.”
Astronomical calculations are not to be relied upon.
The basic principle with regard to sighting the new moon is that it should be seen with the naked eye, but if the new moon is seen with the aid of modern instruments, then this sighting may be acted upon, as explained previously in the answer to question no. 106489.
If the new moon is sighted in one of the states of the US that is close to your state and the moon could be seen there at the same time as in your state, then you must fast with them. The majority of scholars are of the view that it is obligatory to fast, even if the moon could be sighted in one country but not another.
It says in al-Mawsoo‘ah al-Fiqhiyyah (23/142): The Hanafis, Maalikis and Hanbalis are of the view – and it is one view among the Shaafa‘is – that it does not matter if the moon in sighted in one country and not another with regard to proving that the month of Ramadan has begun. If the sighting of the new moon of Ramadan is proven in one country, then all the Muslims in all countries must fast. That is because the Prophet (blessings and peace of Allah be upon him) said: “Fast when you see it,” and this is addressed to the entire ummah.
The more correct view according to the Shaafa‘is is that if the moon can be sighted in one country but not another, this is something to be taken into consideration (i.e., in the country that is so far away from the country where the moon is sighted that the moon cannot be sighted there), then the start of Ramadan must be different in those two countries. End quote.
The view that is more likely to be correct is that if the moon can be sighted in one country but not another, then it sufficient to determine that Ramadan has begun everywhere.
For more information, please see the answers to questions no. 50487 and 1248.
If news of the sighting of the new moon reached you after Fajr, and you had not intended to fast from the night before, then you must stop eating and drinking for the rest of the day, but you have to make up for that day according to the majority of scholars.
Ibn Qudaamah (may Allah have mercy on him) said: If someone woke up not fasting, believing that it was still Sha‘baan, then proof was established that the moon had been sighted, he has to stop eating and drinking, but he must make up that day, according to the view of most of the fuqaha’.
End quote from al-Mughni (3/34).
See also the answer to question no. 205789.
But if you went to sleep with the intention “If tomorrow is Ramadan, then I will fast,” meaning that if it was proven that Ramadan had begun, then your fast is valid and you are not required to make it up, according to the more correct scholarly view. This is the view of the Hanafis, and is mentioned in one report from Ahmad, as has been explained previously in the answer to question no. 70479.
And Allah knows best.