Yes Dave, I agree with all here, just put them in a container and put that in your cave, adjust the lid to get the right humidity. The young cheese still drips and sweat so much moisture, you will have no problem getting 95% humidity.
As far as drying them at such high humidity, yes, it does sound like an oxymoron, but remember that relative humidity is the amount of water vapor in the air relative to the maximum vapor that air can hold -not air that's 100% water. If air at that temperature and atmospheric pressure can hold 1% water vapor before the vapor turns into water droplets or condensation, 95% RH is really 95% of that 1%. That's still more than 99% air - so the cheese will dry.
This is a lot like cheese brine. If you ever made saturated brine you know that water get saturated with 26.5% salt maximum. If you keep pouring salt into it you will find salt crystals on the bottom of the container which will not dissolve because the 100% salt saturation point has been exceeded. Same thing with air in a box. Once you reach 100% saturation, any excess vapor just turns to droplets on the box.
So, a perfect trick to know if you are at about 95% RH without using a hygrometer is to look at the box. If you see tiny water droplets sticking to the lid and they are just shy of being large enough to form drops, then you are at the right place. Once you practice that enough you will easily be able to guesstimate how much to open or close a lid at any day, for any cheese!