Adiabatic expansion would contribute to some of the heat loss but you're right that most of it is from evaporation - compressed air cans aren't literally atmospheric air that's been compressed.
I mean you shouldn’t be holding the sprayer down no matter what. You shouldn’t be doing short bursts and moving it quickly like a spray paint can. And when it gets cold shake it a bunch and wait for it to heat up. If you can see the refrigerant coming out then it’s too cold and you need to stop.
Just be careful not to aim it at capacitors because they might clip off. Also just wait for the bottle to warm up when it gets cold, don't remember why, I just remember it says on the can
It's been a while since thermodynamics, but it's because of adiabatic expansion of the air. Basically the same thing happens when you blow, the air becomes cooler when it expands outside your mouth