I generally ship 2-3 boxes of cookies a month to overseas military folk I know.
It's a great excuse to keep the burnt and the broken for the household...
The least expensive route in terms of shipping costs is the Flat Rate Priority boxes from the Post Office. They are boxes you get from them for free and you can pack as little or as much into them for one rate. The weight limit is something like 20 pounds, but even in the biggest one, I've never managed to come close to the amount shipping cookies.
It costs 10.95 to ship the largest box to the APO/FPO addresses, but that's discounted for the troops. I think the largest box is 12.95 or so stateside.
I do ship holiday boxes to family stateside - and again, I use the Flat Rate Priority Boxes - they generally get where they are going in 2-3 days.
As for packing, I use the largest box available and pack cookies in gallon ziplocs, I can stack 3 doz in each. Plus a piece of bread during the hot months (Apr-Oct). I put some newspaper (the funnies and good stuff to read) in the bottom and top, not squished, just flat. Then I pack 3 bags into them. I've got enough room left over for a bag of coffee (send ground) and the little extra space I cram with more newspaper, folded.
I generally just do chocolate chip, oatmeal spice and similar cookies. It takes anywhere from 4 days to 2 weeks to get to them, so I look for durable cookies that will handle the journey. Heat is the biggest factor for shipping to troops in the Middle East. Of course with them, they are just so happy to get them, they don't care if they're a bit stale or crumpled. They share with everyone and the cookies last about 2-3 hours, sometimes less.
For more fragile cookies I go with the semi-disposable plastic containers available in bulk at the grocery store, and pack them snugly inside. The less shifting the cookies do, the less likely they are to arrive in bits and pieces. Then I pack the small containers inside the box very tightly, again to lessen the shifting.
At this time of year, the cookies getting moldy are not so much of an issue and the opposite, getting dry and stale shouldn't happen as they aren't in transit that long nor do they get really hot, like over 80-90 degrees.