[shilala]

Andrew, sap from spanish cedar will not only drip, but it'll vaporize and then settle on anything and everything. It'll make hinges stick, lids seal shut, trays glue to glides, rollers stop rolling, and all kinds of crazy stuff.
The only way I could imagine this happening the way that it is if the cigars are colder than the air surrounding them, and the volatiles are condensing each time you open your humi.
If it were happening to me, I'd obviously have more info to go on, but I'd make sure all my smokes were in cabs or boxes, closed tightly.
I'd probably empty the humidor and let all the wood come up to room temp, or maybe put a low wattage lightbulb in there to warm the wood and draw the sap out to see if that's really the problem.
Being that you mentioned it's on a number of cigars and not just those, I'm guessing you store in open trays and not in cabs or dress boxes?

Cedar pitch is different than pine pitch. It's very sticky, but it tries real hard to dry out, whereas pine kind of stays real gooey and pliable and smeary. They're different colors, too. Cedar pitch is darker.

Mind ya, I'm just guessing. I don't have remotely enough info from you to help much, I'm just relaying my experience with the wood as I've used it, or actually as I've removed it and replaced it with stuff that doesn't weep.
I'm also guessing that the sticks are refrigerated and in open trays, it just looks that way to me.
I sure hope you get this ironed out!!!