One can tell a New Moon by oneself if one has been taught how, but the more complex sides of the calendar are not so easy to work out.David was hiding from the King.
No ways he was going to search for a Priest to tell him that "tomorrow" is the New Moon.
And as you know the moon is not visible until it is declared/sighted as a sliver on more or less the 3rd day after conjunction.
So, respectfully I say you should go speak to your rabbi about this.