Bit of a clart but maybe use vacuum bags to seal items before they go in the box?
Very tricky with the Really Useful boxes as the lid is not flat all the way around when it meets the box so therefore you cannot add it all the way around for a secure seal I've had a look at some and some say yes and some say no. The problem with damp is that they need a silicon or some other kind of seal to stop air getting in. The Really Useful ones are great but there's a lot of air flow.
the really useful box on amazon have amazing reviews saying damp proof
Get some of this to make an airtight seal and put some boil in the bag rice in to absorb any residual moisture in the product
Look in your local homebase as they scan at a similar price