In Texas and other humid places it's probably a must and I would budget adding it in 5-10K based on property size.

I would imagine that bug/rodent control and problems related to humidity will end up costing more, but how much more should I plan for?

Mold needs oxygen and humidity/moisture to survive, so if you get the moisture issues fixed and rob it of oxygen it won't come back.

Then the humidity got crazy high in the house, and the paint started coming off the walls.

If they get the slightest bit wet they will discolor and look awful, and if there is ever a humidity issue those tiles will hold a musty odor forever.

you have to reduce the humidity level in the house.

So I'd check with city code book and see what's required,, all that humidity has to go somewhere,, and inside isn't where it belongs.Even if it was a private home,, the amount of moisture staying inside is bad for your house structure,, mold issues, paint peeling, rot, over time it will become a bigger problem

Remember you are in a humid climate so you vapor barrier wants to be on the outside not the interior side.

They said the mold in the garage (basement) that was on tenants shoes and clothing was due to the humid conditions and rain.

Now I figure there may be a humidity issue in the basement, but does that warrant the dehumidifier running 24/7?