I used to order the mugs directly shipped to me and then send them out from my home. It was a ton of extra work but i figured it has to be cheaper to order a bulk amount of mugs.
However with shipping prices in the US through the roof, it turns out to be cheaper (and a ton less work) to let CafePress drop ship them from china or wherever, individually to each person.
This is my way of saying that while you might think you can come up with a cheaper way to make and ship them by doing a lot of work, it might simply be the case that an online site like cafepress or whatever has just optimized things well enough that they do a cheaper job of sending things directly from the factory to different people.
As for a shirt, i think it's a great idea.. but something to consider. If there are 30 entries in the NANY, it means we spend about $600 on mugs each year (about $20 each; often more because people who participate in NANY multiple years can ask for something more elaborate than a mug). Even if you got the tshirts to half price including shipping, it's a lot of money.
HOWEVER, I think a really nice NANY shirt design would be fantastic, and might be something people would prefer over a mug, or might be willing to buy themselves at cost in addition to the mug.
So my advice is, don't worry about sending out free shirts -- instead concentrate all efforts on having a reallllllly cool new style NANY/CODY shirt made that people would LOVE to have.. Something really special. Maybe it doesnt have to be NANY specific, but an awesome CODY DC shirt.. The plain white tshirts with a picture of cody just don't cut it. We need something really modern and nice I think. Then we can worry about how people get it later.