I think Dan might be on track, in that removing the moisture might help.
Your only other option would be to use some kind of coating/varnish to protect it from the environment. The problem is that unless you gently and carefully first heated the entire structure before you coated it to remove the moisture, you would just be locking in the already existing moisture. That might not degrade as fast, or all over, but there will still be major problem spots. Using the desiccant, if it works, you could always refill it.
Also, make sure the room you keep it in is not to humid.
However, glucose reacts with oxygen to produce carbon dioxide, water and heat. I do not have my reference material with me right here, so I can not look up the energy required for that reaction to proceed, and I do not have the time right now to scour the internet for it; perhaps someone else might know/can look it up real quick. If it is low enough and this might be the problem, the easiest solution would be to refrigerate it when not on display (say, over night) in the desiccant.