I think that a lot of it depends on climate and humidity where you are. Its a matter of trial and error for all of us.
Here it is very cold and dry in the winter - in WA quite a bit warmer and rainy I suspect.
Your green house is probably getting pretty warm still when you are digging tubers so thats probably why they are drying out so fast. The only time I have had tubers shrivel before storing is when they were laying out for 2 weeks before I go them all divided and wrapped. Peat moss or vermiculite is going to hold the moisture better than paper or shavings do, But you live in a pretty humid part of the country. What kind of container are you storing them in?
Because I individually wrap mine in saran, I use cardboard boxes, or if I use a plastic bin, it doesn't get closed tightly, but packing them in loose dry material - I would think a closed plastic bin would be better.
If you have the space, storing whole tuber clumps still covered in dirt works well for lots of people. And in your part of the world (if you have well draining soil) you could potentially leave them in the ground over-winter, and dig and divide in the spring couldn't you?
