What can I use to quickly color already-folded origami?
An airbrush seems perfectly suited to this type of work. You'd be able to adjust the flow and target the crevices, to get even coverage without soaking the piece.

Unfortunately, the start-up costs of getting an airbrush are considerably more expensive than grabbing a can of spray paint or using some sort of dry brush technique.

There are tutorials on how to make an airbrush available online. I won't cover them here, because that's a bit beyond my scope. It is an option if you're on a budget.

Renting an airbrush might not be an option, but there may be studios in your area that let you use their equipment if you pay a cover charge and for supplies. I know many studios, especially on college campuses, allow you to use studios for about $10 and it gives you access to all their equipment, generally pottery wheels and kilns and tools. If this is something you'd want to pursue, I'd call around and see if they have an airbrush available if you pay for some studio time. Then, make sure you bring enough to do or have some other ideas to explore, and make good use of that time!