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Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahrzeits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah. In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations.
This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years.
These three phenomena are independent of each other, so there is no direct correlation between them.
On average, the moon revolves around the Earth in about 29½ days.
To compensate for this drift, the Jewish calendar uses a 12-month lunar calendar with an extra month occasionally added.
The month of Nissan occurs 11 days earlier each year for two or three years, and then jumps forward 30 days, balancing out the drift.
If you are interested in the details of how these calculations are performed, see The Jewish Calendar: A Closer Look. This concept of different starting points for a year is not as strange as it might seem at first glance.Holidays are celebrated on the same day of the Jewish calendar every year, but the Jewish year is not the same length as a solar year on the civil calendar used by most of the western world, so the date shifts on the civil calendar.