First, I would recommend taking your cigars out of the wine cooler and putting them in a Tupperware or similar container or Ziploc bags along with the beads. Cigars are resilient and will be fine.
Empty, your wine cooler should hold a steady RH for at least 48 hours. If you place a wet sponge in your empty wine cooler with a hygrometer, raise the RH until it holds steady (at least a day) at say, 70% (or even more), then remove the sponge, the RH should stay steady. Even when the compressor comes on, the RH should raise back up when the compressor turns off. If you have a lot of wood, it may take a bit longer for the RH to rise to your desired level.
First, make sure that the RH holds steady without the compressor running. If that's OK, then turn the compressor on. You should measure the recovery time (how long it takes the RH to rise back up), and if it does not come back up, then look for other issues.
If the RH did not hold steady with the compressor off, then we need to look at other things first.
Have you fully seasoned the shelves/drawers? Any (and all) wood in a humidor needs to be seasoned periodically; at least twice a year is good.
Also, have you checked the door seal? Put some Vaseline (or similar gel) all around the door-side seal -- a complete as-smooth-as you-can bead. Then close the door and open it. Look for any gaps in the Vaseline on the cooler-side of the frame and those indicate a leak. (note: do not do this on wooden humidors!)
You only get one good shot at this -- you have to completely wipe the Vaseline off before doing it a second time.
Sorry if this is rambling... it's late
