Hi again, you seem to be going down the same route I have taken. I use Plano 3700 series DEEP PROLATCH STOWAWAY 4-15 adjustable compartments.
14"L x 9.13"W x 3.25"H
Why I like these? Inexpensive, provides a great seal where you can put a Bovedo pouch in that controls humidity both ways. If you want you can also put foam in the compartments to take up the space but I do not do that. I just handle them with care. I ordered mine from Amazon Prime. I also keep the original packaging of the chess set if it comes with form fitted foam in case I would have to transport them one day which I hope I never have to do.
I did find a cigar box at Goodwill which can also be used and looks fancy because of the nice finish but as a practical box I never have found anything in the price range to beat the Plano.
Jut make sure you do not order a box that is too short in height. I do see some now that are totally waterproof sealed with o-rings but haven't taken a look at the dimensions.
This is he exact one O have been ordering ut you my find a different size better for you.
https://www.amazon.com/Plano-23730-05-Prolatch-Stowaway-Adjustable/dp/B001447AW2/ref=sr_1_3?s=sporting-goods&ie=UTF8&qid=1519749714&sr=1-3&keywords=plano+2-3730-05
Information......just last year the price of these were $6.57 , I bought a few. I see the price has increased. Still worth the price I feel.
I've heard that some people use these clear plastic boxes to store their sets. Does one box hold the entire set, or just the white or black pieces? Are these boxes working well for anyone?