Did the new moon have any religious significance in itself in the Old Testament, or only in connection with sacrifices?

Anything that revolves has a beginning and an ending point, so you had the sun with its solar cycle and the moon with its roughly twenty-eight-day cycle, giving about nineteen cycles in a year. Because these cycles were so precise, you could plan a feast to fall at the same time as the new moon; you knew when the new moon was coming, so you could say, we will hold this feast in seven days when the new moon arrives. That let them keep a calendar for their feast days and Sabbaths. Remember, God built the whole creation on seven days, and in Exodus 20 He tells the people to work six days and rest the seventh because He worked six days in creation and rested. That paradigm of doing certain things on certain days in certain cycles is built right into creation, even the twenty-four-hour rhythm of evening and morning in Genesis 1, which biology calls the circadian rhythm. So the new moon and all these cycles were important, but once they became tied to worship, they became part of the ceremonial law of Israel. Anything to do with ceremonial worship, like the seventh-day Sabbath, was done away with when Jesus came. So the new moons and Sabbath cycles of the Old Testament are all abolished; we no longer tie our worship to those time sequences. Now it is simple: we worship on the first day of the week. The Church created feast days for the saints, but those are not obligatory and are not tied to our worship; they simply help us in daily life.