Shipping ammo is no big deal other than the cost due to weight. UPS is your only option at this time. The outside of the box must be marked ORMD Small Arms Ammunition.
I just sold @ 10 boxes of 22-250. I went to our local everything store and asked for a box to ship ammo. They gave me a Remington box that had the appropriate markings. UPS required an adault signature wich adds a little more to the shipping cost.
Over all no big deal.
There is a big difference in the cost of the loads you have. Check the boxes and get the exact load info. The premium loads with Barnes and Swift bullets are very pricey. If they are just Nosler then they are not as expensive. Also, Some of the older Weatherby boxes are worth as much as the ammo alone. So if you have old boxes don't trash them before checking. If you have unloaded brass for some of the bigger stuff it is worth quite a bit.