I will note that the book of Enoch, Chapter 77, which is not scripture, and Philo who lived in Alexandria during YeHoVaH's day, seem to indicate the month starts with the dark conjunction. Enoch could actually be interpreted as a first-sliver start of month, while Philo clearly indicates that the 15th of the month must be the Full Moon.

On the fifteenth day, at full moon, the feast which is called "the feast of booths" is celebrated - Philo

It is noteworthy that Philo appears to have been a Pharisee; therefore, while his opinion supports the Lunar Sabbath, it could be wrong about following the dark conjunction.

The Pharisees lacked the political power of the Sadducees who ran the Temple; therefore, they did not get to set the Temple calendar. The question remains whether Pharisees would practice something different than they preach or go along with a potentially different calendar run by the Sadducees. Given the death penalty or similar harsh punishment for violating the law of the Sadducees I suspect everyone would obey the opinion of the Sadducees. Regrettably, I don't have similar texts to support the Full Moon.

Following the order which we have adopted, we proceed to speak of the third festival, that of the new moon. First of all, because it is the beginning of the month, and the beginning, whether of number or of time, is honourable. Secondly, because at this time there is nothing in the whole of heaven destitute of light. (141) Thirdly, because at that period the more powerful and important body gives a portion of necessary assistance to the less important and weaker body; for, at the time of the new moon, the sun begins to illuminate the moon with a light which is visible to the outward senses, and then she displays her own beauty to the beholders. - Philo

This gives rise to interpreting Psalm 81 as blowing the trumpet at the New Moon, and on the Full Moon of the Feast.