Keeping my birds cool is a much bigger challenge than I thought it would be. We don't have a downstairs air conditioner, so the living room temperature often gets into the upper 80s, even with the bedroom one running and the doors open. My birds will sit quietly with their wings slightly out when they are hot. I have tried misting them with a spray bottle, but they always freak out and have never gotten used to it. I have also tried supplying a small bowl of water for them to take baths in since their water dish seems a little small for that and they ought to be drinking that water, not getting it dirty. They are always afraid of the new object thought and don't even touch it. I am not sure why that is since they are supposed to be naturally curious.
What do you do to keep your birds cool and comfortable in the summer?
You can use a fan -- not directed at your birds. The breeze created by the circulated air will help to cool them down. If your birds are outside the cage never turn on a ceiling fan, unless the wings are clipped to prevent flying into the fan.
You stated your birds don't like being misted. This is a common reaction, but some birds do enjoy it. I know mine did.
Keep the cage out of direct sunlight, sheltered from the sun. If your cage is near a window, remember the sun changes direction during daytime, so make sure your birds can get into the shade while caged.