I used to ship and receive tons of corals in a "dry" state but you need to pay attention to temperature especially this time of year. Even a styrofoam or insulated box does not maintain temperatures well if it is full of mostly air. You also do not want to ship corals "dry" with pure oxygen as it will burn the tissue so just fill with air. With a small amount of water, enough to cover the bottom of the bag, loosely wrap the coral with another plastic bag and set it in the bottom. Always triple bag with at least 4mil plastic and pad around the coral so it does not bounce around in the insulated box. Balled up newspaper works very well to fill any voids in the box. Also always pay for overnight shipping so heat (or cold) is not as big of an issue and the least amount of time in the bag the healthier and less traumatized the animal will be. Depending on outside temperatures in your area an ice pack may also be a good idea but never let an ice pack or heat pack for that matter come in direct contact with the bag the live coral is packed in.
You can always do a test package at your house packed like you are going to ship, let it set out and check it for temperature after 12 to 14 hours (without live animals in it of course). If using Fed Ex last drop or pickup time is 7pm and you can still get guaranteed arrival by noon the next day just about anywhere in the US.