The hotel has a bar in the basement where you can buy liquor but if you want wholesale liquor go to downtown spirits or search "downtown spirits seattle"
But the Fairmont is a full service hotel. Give them your order and they can have it delivered while your there.
There's a Bartell Drug Store a block or so away that has beer, wine, and liquor.
Yes about 100 yards south across the junction just below the hotel entrance and part of the grocery store.
Get an answer from The Fairmont Olympic Seattle staff and past guests.