What month is the moon? I was born on the second day of October in the lunar calendar, but the eight characters count as the full moon. What month should it be?

The eight-character lunar calendar is calculated according to solar terms. It depends on which day you were in the early winter of that year. You said you were on the second day of the tenth lunar month. If you were born in beginning of winter in that year 10/0, you were born before beginning of winter, so it is the moon (September). If you were born in beginning of winter in 1 year 10, and you were born after beginning of winter, it is the moon (10 June).