I think that you already know the answer that you suspect: high humidity.
The level of risk that it represents is probably best gaged by you wearing a pair of eyeglasses .... under those conditions that they fog up when you enter the indoor pool area, the same thing is potentially happening to the camera gear.
Basically, carrying a "cold" camera into a 'warmer" indoor pool room is where the big water precipitation risk is.
It seems that it would be straightforward in that you could avoid this is problem by "baking" the camera before taking it into the humid room, such as by blowing a hairdryer on it.
Yes, this does work, but as the camera cools down to the indoor pool's room temperature, the cooling of the previously 'warm air' inside the camera will shrink, and suck the humid air into the camera body, which could cause mischief...there is no free lunch.
However, that's still probably your better bet, unless you're going to go with a full waterproof (scuba) case. You can also re-"hairdryer" the camera after the exposure, to reheat it up and drive the water back out, whereupon the air that it sucks in after it cools back down again will be from the lower humidty room that you used for this post-exposure period.
FWIW, what can be useful for any of this is to consider putting your camera in a sealed ziplock bag with some anti-dessicant packs...help to keep it dried out. You can also put the ziplocked-camera into the warm/humid indoor pool room to thermally acclimate fairly quickly - - leaving it in your camera bag can also work, but because such bags often have a lot of padding, they generally represent some insulation "R" value too, so thermal acclimation takes longer.
FYI, if you have a ziplock bag but no anti-dessicant packs, you can zap the air in the bag with a hairdryer for a few seconds...this helps reduce the humidity in the bag before you seal it.
This same basic technique is used by (some) underwater photographers before we seal our cameras inside of UW housings: it dehumidifies the air trapped inside with the camera, which lowers the dewpoint for when water condensation could form on the inside of the camera's dome port, which can happen when the system is submerged in cold water.
-hh