The more you handle the dough after cutting in the butter, the less sandy the texture will be. This happens because of the temperature of your hands! Also, with handling, the butter works further into the flour, which then gives the shortbread a more cakelike consistency.
Walker's Shortbread is 31% butter, 18% sugar, and just over 50% flour, with a little added salt. The closest I have come to Walker's shortbread is the following recipe:
3 c flour; 5/8 c sugar; 1/2 tsp salt, 2 sticks butter (at room temperature)
Thoroughly mix dry ingredients. using two forks, cut in butter until mixture resembles coarse meal. This will take some time, so be patient.
NOTE: Store the shortbread at room temperature, in a well-sealed tin or bowl. Homemade shortbread will last quite a while, if properly stored. I occasionally buy small holiday tins at Christmastime, and package the shortbread for gifts. If you do this, make SURE to separate the cookies with paper, and don't stack the pieces over three high.
Oh, and just as an interesting aside, I think every restaurant I've ever eaten at in France uses this glassware! I first encountered them in a brasserie in Cannes, back in the sixties, and on my last trip to France, I had dinner at the Le Cognette in Issodun, (which has a nifty collection of Michelin Stars and charges accordingly!). Guess what I drank from?!?
Rebekah, if you're still looking for Duralex glasses in the vintage pattern, I have three in each size from my father-in-law's estate. They seem to be selling (used) for around $6 each. Make me an offer and pay shipping, and they're yours.