I have had lots of trouble with these guys, and I will tell you what I discovered (not that I'm an expert by any means)!

Carpenter ants live in dead/rotting wood, and build their nests. They do not kill trees or attack fresh, healthy wood, but set up nests in dead wood.

Going back over 25 year period, every time I had an onslaught of carpenter ants invade my home, there was ALWAYS dead wood in the immediate vicinity. As soon as I found the dead tree/dead limb/ rotting stump/ etc. etc. the ants disappeared in one day (no kidding!).

It's possible they may have formed a nest inside your house, but I would carefully check a perimeter up to 60 feet radius around your home. I found them once by going out after dark with a powerful flashlight, and , sure enough, there was a two-lane line of them going back and forth from a big elm 50 ft. away to my house, up the foundation and under the siding. I sprinkled diazonin around the base of the tree and that stopped it in one day. I got a tree man out, and my tree had Dutch Elm disease, and actually was dying. The ants were in a dead limb 50 ft. up in the air. I had the tree cut down, and no more problems. When the men cut the dead limb, thousands of carpenter ants ran in every direction.
I have had several experiences like that. Once my mother was getting them in her house, and I discovered an old plastic bucket overturned on her rear patio. There were hundreds under it, and had formed a colony under the bucket!
They were gone the same day.
Anyway, I could go on with other experiences, but it's always the same. They are coming from outside the house.
Hope this helps!

Sealing all cracks and crevices and possible entries into the home will keep insect and rodent pests at bay. A regular perimeter insecticide treatment around the foundation of your home will make a significant impact on invasive insect pests.

Make sure all gutters and downspouts are clear and excess water directed away from the foundation of your home. If gutters are clogged, water could be directed back into the walls of your home. Improperly installed vinyl siding could be allowing moisture into the walls of the home. Cut overgrown trees and shrubs around your home to all adequate ventilation. Improperly installed vapor barriers and insulation in the walls could be causing condensation problems that will cause moisture problems, decay, and mold/mildew inside your walls. Mold/mildew can cause allergies, respiratory problems, and other health problems. It might be worth it to have your home inspected by a professional inspector.

if the drywall is damp, isn't it possible the wood in the house is damp, and rotting somewhere, and so the carpenter ants could build a nest in the house?

Look at the roof, when they replaced the roof on the house I was renting, most of the boards on the back side were rotted and infested with ants. After all the boards were replaced, I had them come out and spray and I have never had another problem 4 years now. Ended up buying the house.
You have to have wet wood for them, find it and you will be able to get rid of them and stop the damage.

Scott

and I've realized your first question, which no one has answered really, is how to go about finding how the moisture is getting in.

First step might be a housing inspector-type person. (do you own? if so, didn't your housing inspector find anything?)

Most obvious person is a roofer. Partly because the roof is the obvious spot, and because even if it isn't, he'll probably have lots of experience trying to figure out the source of leaks when it isn't the roof (because those people will have called him first).

Then, if the roofer strikes out, a siding guy.

The roof and the siding are your home's raincoat. Check them out first.

Plumbing problems would show up in your water bill, I'd think, but if roof and siding don't have vulnerabilities, then plumbing's next.

Thanks Talley sue-great advice!