Depends on how dorky you're willing to look in shorts.
The best thing I've ever used to prevent shown splints -- and I get them a lot -- are compression sleeves on my calves. Shin splits are actually caused by micro tears in the muscles on the shin and are caused by them jiggling more than they are used to. The sleeves keep the muscles from sliding, and cuts out the tears.
Those are hard muscles to stretch out, because there just aren't many of them on the front of the leg, and from what I've been told, getting them is as much genetic as it is conditioning. (conditioning helps, don't get me wrong, but because there's not a lot of muscle there, it's hard to really develop more). You are probably walking faster and with longer strides at WDW than at home, and that adds to the tearing.
So I'd hit a sports or running store, and get a pair. It's no big deal under jeans or long pants, but in shorts, you'll look pretty dorky. But your legs won't hurt as much.