Yes, the Sun changes sidereal signs pretty much every year on the same day.
I don't have all the answers to your questions, as I'm only at the beginning of this research. I'm still looking into what causes or prevents the larger cycles, why one year yes, and the other no, but I hardly made any progress there.
However, I found that many peaks occur when the Sun changes signs. Also, often cycles start then. Another way to convince yourself is keeping a detailed record of peaks and then comparing it to solar ingresses by using an ephemeris. If you wish to know more, best take a look the pdf attached, as it includes my findings up till now. For your convenience, I've listed the solar ingresses for the rest of the year.