In the scriptures, is not the first day of a month when there is a new moon?

II Samuel 20:24 ¶ So David hid himself in the field: and when the new moon was come, the king sat him down to eat meat.

  25 And the king sat upon his seat, as at other times, even upon a seat by the wall: and Jonathan arose, and Abner sat by Saul’s side, and David’s place was empty.

  26 Nevertheless Saul spake not any thing that day: for he thought, Something hath befallen him, he is not clean; surely he is not clean.

  27 And it came to pass on the morrow, which was the second day of the month, that David’s place was empty: and Saul said unto Jonathan his son, Wherefore cometh not the son of Jesse to meat, neither yesterday, nor to day?

This, therefore, can happen on any day of the week?

If so, then is not the fifteenth day of the month when there is a full moon?

Leviticus 23:6 And on the fifteenth day of the same month is the feast of unleavened bread unto the LORD: seven days ye must eat unleavened bread.

  34 Speak unto the children of Israel, saying, The fifteenth day of this seventh month shall be the feast of tabernacles for seven days unto the LORD.

So, the feast of tabernacles cannot begin at another time, in other words, it cannot begin when the moon is not full?