I'm glad you're finally going to get a darkroom you'll be able to enjoy for years to come. I wish you happy developing and printing!
My husband (bless him) built my darkroom in a shed in our garden. The wet side is on one of the two longer walls, the dry side on the other. The enlarger is in between on the far (shorter) wall - the trough is a couple of feet away from the enlarger.
He used Ikea kitchen cabinet frames under the countertops (cheap, cheerful, last a lifetime). Two of the cabinets have doors. In one, under the dry side, I keep my developed negatives and prints, stored in Really Useful Boxes (that's the brand name) for added protection. They are designed to live in basements so offer reasonable protection from the elements.
In the other, under the enlarger, I keep the paper. I keep my 4x5 negatives in an old ammo box, which is designed to keep the weather out.
I also have a small camping heater (the kind that cuts out if it falls over) to heat the shed in winter and reduce the humidity a little (it rains A LOT in England). I run the heater on low for an hour a day, plus it's on when I'm working in the darkroom. Since it's a little heater, it uses hardly any electricity.
So far I've had no problems with damp. The paper is fine, the negatives are fine.
Bruce Barnbaum, one of my favourite B&W photographer/printers, has kept his negatives and papers in his darkroom for years and they're fine.