Exodus 12 does speak of the Passover celebration and 12.2 reads, "This month shall mark for you the beginning of months; it shall be the first month of the year for you." But, as indicated, Exodus 23 clearly states the end of the year is in the fall, after the harvest, "You shall observe the festival of ingathering at the end of the year, when you gather in from the field the fruit of your labor" (Exodus 23:16).
In answer to the question, is the Biblical New Year in the spring or fall, the answer is-Yes.
To understand why Exodus, and indeed Leviticus, Numbers, and Deuteronomy, contradicts its own New Year's beginning, we have to understand how we got the Torah, that is, the Books of the Law.
It is clear the five books of Moses weren't written by Moses at all. Though he is the prime character through four of the five books, he didn't pen the works. Instead, the stories and the accounts existed only in the oral medium of the early Israelites. These were passed on between generations mainly through storytelling. As time passed some of these stories were recorded-some in Judah and the Southern Kingdom, others in Samaria and the Northern Kingdom.
In 587 BC Judah fell to the Babylonians. Jerusalem and the Temple were destroyed and the wealthy, powerful, and learned were deported to the north. Without the Temple the priests and religious leaders had time on their hands. They began compiling the stories and accounts of their history and began to codify them (put them into a written systematic form).
However, as mentioned above, some stories came from the Northern Kingdom. Others came from the Southern Kingdom. And some accounts were quite ancient while others were rather contemporary. Apparently it was decided to allow the stories and the accounts to stand for themselves. Though the religious leaders did a fair bit of editing, getting all the stories to agree and/or trying to make similar stories coalesce wasn't a priority. That's why today we have a variety of views and stories that don't always correspond to each other-the side-by-side creation stories of Genesis 1 and 2 for instance.
Now, to answer our question more specifically. Exodus was "written" by at least three hands and over a long period of time. Early in Israelite history, the calendar followed the agricultural cycle. The year began with the rainy season, just after the harvest. Exodus 23 best illuminates this period of Israelite history. In verses 14-16 the Israelites are commanded to observe three festivals each year-the Feast of Unleavened Bread in the spring, with no mention of Passover; the Festival of Harvest, a festival of the first harvest; and the Festival of Ingathering, the festival at the end of the year commemorating the gathering of the harvest.
In later years, likely during the monarchy, these festivals took on additional meanings and traditions. Passover was added to the first day of the Feast of Unleavened Bread, in the spring, and the date of the New Year was changed to reflect the importance of this holy-day. These new traditions are recorded in Exodus 12 and in Leviticus 23.
Today, Rosh Hashannah in the autumn marks the beginning of the Jewish year, in accordance to the most ancient of the Torah's instructions. But the Biblical New Year remains enigmatic-it's a festival in the spring and/or the fall.