I actually found that I have a medium sized cooler that I forgot about and I think it will work great to hold my sticks until the drawers are ready.
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Now that I had a second to re-read your question...
I only seasoned mine for 3 days before i became impatient. I think if you just throw your cigars in a cooler in the meantime you should be fine. but a few other BOTLs said "just load her up and let it season over time." I don't know if i fully agree with that advice but it sure didn't change the outcome for me. since my drawers were still sucking up moisture for a few weeks. i just noticed that I had to recharge my HF Beads every few days to a week but now it is solid and I haven't had to recharge in over a month i think.
Most cedar comes kiln dried, so basically zero humidity. Most places the ambient humidity is 30-50%. So you could just have the drawers out in the open for a week or two and they'd naturally start absorbing some humidity. Then if you were to just drop them in the wineador with the cigars the relative difference would be less.
I made the massive mistake of seasoning my drawers when I got them. I was stuck at 80% rh for over a week. Seasoning is definitely not required in Florida. I actually never have to recharge my beads. On the contrary...I actually have to nuke them every few months to remove the humidity. So results will vary a little to a lot depending on geographic location.
Then the second huge mistake was that I thought my collection would fit in a single 28-bottle wineador. I was mistaken by about 300 sticks. A year later and I still have overflow in the old coolidor. So far I have managed to resist the urge to build a 2nd one but its inevitable :D