Seems your linked website as a very believable conclusion attached to it. I have a similar issue where the relay module would behave erratically. In my case it was when the relay module/595 chips received power before the Arduino was fully powered up. It’s unlikely that a load affects your relay module as they should never be connected to the main circuit.
I guess something like a pulldown resistor on the SER, CLK and RCLK pins would solve the issue since that would kill any noise. The noise is probably the kind of voltage that BARELY registers as HIGH but highly random.
That or making the relay module only turn on after the microcontroller has finished starting up using a mosfet or something.