When I lived in Arizona I used a swamp cooler in a sunroom and it did not put enough moisture into the room to dampen anything other than the water it spit out on the floor. The Arizona air is so dry it sucks the moisture out of everything so fast nothing ever stays damp. I made the mistake of taking the same swamp cooler to Tenneessee when we moved and used it once! Moisture was dripping down the walls...what a mess!
I think you will be fine, but keep an eye on things. If things get damp turn it off.