Why Does the Saju Month Change at a Solar Term, Not on the 1st?

Because in Saju, a month begins and ends at a solar term. In Saju theory the boundary of a month is neither the calendar's 1st nor the lunar new moon but the solar term — so anyone reading the flow uses the same ruler. The birthday-calculation side of this lives in the lunar vs. solar calendar note; here we only look at the unit of time that fortune is cut by.

How does the Saju month differ from the calendar month?

A calendar month turns on the 1st; a Saju month turns at the solar-term entry. Solar terms are the seasonal joints that divide the sun's path into twenty-four, and the entry moment is when that term's energy arrives. You don't cross into the Myo month until the entry of Gyeongchip (the Awakening of Insects), for instance — so the same calendar date can carry a different Month Pillar depending on whether the term has passed. The Ten-Thousand-Year calendar records these boundaries precisely, down to the minute.

What changes when the next solar term arrives?

What changes is that month's stem-branch pair — the characters of its energy. Just as today's fortune reads the day's stem-branch, monthly fortune is the month-long grain made where that month's pair meets your own chart. But note what that claim actually is: the characters changing is a fact of calendar calculation, and nothing in it says next month gets better or worse. Reading this rhythm, too, is not a fixed fate — just a reference for looking back over a month.