I've never used the beads myself, but they sound like chemically tweaked silicone packs. I would be willing to bet money that if you put in a decent quantity of dry ones, they'll suck up a lot of moisture. I also bet you can dry them out the same way you do silicone packs for gun safes- put them in the oven @ ~250 degrees for a few hours. When they come out, they'll be ready to suck up tons more moisture. This oven bit may be overkill- the beads may soak up too much when they're that dry. Try just using beads that are at room RH, then if you need more drying, use the oven trick.
This all may be moot if the idea about removing one humidifier works. Try the simplest solutions first!
I've tried opening the humi to air out. It works, but seems like a poor long term solution to me. That really exposes your cigars to severe ups and downs.
It sounds like you just have a humidity source that is over exuberant. Are you using Propylene Glycol? This chemically limits how much moisture your humidifier will give off.
Good luck!