The cabinet grade plywood you have is fine, no need to seal the birch.
Just my opinion, but I think we worry sometimes way to much about the humidity in our humidors. Here's my rational, in my home during the summer, it can get much higher than the 65%RH I keep my humidor at, during the winter, considerably less. The fluctuation you have in your house is much greater than what will be inside your relatively stable humidor. Now that being said, you'll have a relative constant RH inside your cabinet and varying RH outside of it, so that's what will give you the greatest chance for failure. With that in mind, using Titebond III wood glue and paying 100% attention to detail in the construction of your cabinet in regard to joint fitment will make all the difference. I don't know what your woodworking skills are, but for me, I mark all my lines with a .5mm mechanical pencil and my goal when I make a cut is to still see half that line on the finished side, if I can leave myself that kind of "receipt" as I call it my accuracy will be right on.
Making that cabinet out of solid Spanish Cedar would not only be way overkill, but also freakin' expensive not to mention less stable than sheet goods. I've seen 3/8 solid Spanish Cedar panels get very moist and swell enough to break the mechanical and chemical joints on a case carcass, yes, even Spanish Cedar moves with moisture change. The plywood idea is ideal, it's much more stable as the grain on each ply is rotated 90 degrees to the ply it's facing, along with adhesives and a heated press to laminate them, they're very stable.
I have pics somewhere on the build process of mine, if I can find the link, I'll post it later.
Take pics, ask as many questions as you need, good luck and have fun.
Wade
