Weird interference when controlling piezoelectric humidifier
I believe the circuit in these Chinese modules uses a boost converter to generate a relatively high voltage for the piezo, in addition to the MOSFET and oscillator. There are two drum inductors on the PCB, so one for the boost converter and one for the piezo. Image below is from this eBay listing but there are many others with the 113kHz frequency listed (not a very atomizing frequency, by the  way).

Boost converters, especially those without a soft-start feature such as the antediluvian MC34063 can draw a massive surge when turned on. This surge, in turn, can interact with current limiting on your power source.
You can try adding a BIG capacitor across the supply, preferably 5,000-10,000uF. Be aware that the capacitor also may interact with the supply and keep it from starting.