@c.goodman,
As long as the buyer has paid for it, then you need to ship it to the address on the order details. If you buy the shipping label through eBay, then it will populate the correct address automatically. If the buyer hasn't paid yet, then you might see their primary shipping address, but that could change when they pay, so you should always until after they have paid.
Your Pokemon card listing does show that it would be in the Authentication program, but the buyer might not be eligible for some reason, in which case you would get the buyer's shipping address and not send it to the authenticators. That can happen if the shipping address is not in the US (or Canada or Australia, IIRC) or if the buyer made the purchase on a different eBay website.
If the buyer is eligible for the authentication, then the label from eBay will be for the authenticator's address.
