Let me know if the applesauce works. I don't see any starches in the recipe. Starches serve to absorb moisture, and glue ingredients together. You might try adding one of the gluten free flours to the recipe before adding the applesauce. If not, then I can see the water from the applesauce flowing all over the cookie sheet and dripping onto the floor of you oven. The following site gives a good list of gluten free grains that can be used in your cookies.
Gluten Free Whole Grains | The Whole Grains Council
The feeling of moistness is governed by several factors as I think about the sensation. The first is extracted liquid when you chew the food. The liquid allows the food to easily move about in the mouth and slide across the tongue. The liquid also carries flavor elements to the taste buds and evaporates into the nasal cavities, enhancing flavors with smells. Another sensation of liquid is viscosity, or a lubricating substance. Another part of the equation is how well the liquid from the food you are eating is made available to the mouth by chewing. Eat a mouthful of crackers and you quickly find that it is nearly impossible to chew. The food just won't move around in your mouth. The same is true of raw flour, or dry, crisped rice cereal. But add milk to the mouth and things begin to move around.
So, if your cookie is dry, that means it can't give up its moisture, either as oil or water, to the mouth. Adding a starch will allow the cookie to hold the moisture until it's squeezed out by chewing. The applesauce helps by holding moisture in the little apple cells distributed through the product. i'm just afraid, and this is only hypothesis on my part, that when your cookie is baked, the water from the applesauce will evaporate, leaving the chewy cellulose behind, like fruit leather. The combination of applesauce and a starchy, gluten-free grain will work together to provide and hold extra moisture in the cookie.
You might also try to find a more moist date. I know that usually the only dates available in supermarkets is the dry, semi-sweet dates. There are wonderfully sweet and moist varieties available, if you look hard enough. Or you might want to try a date filling product instead of whole or pitted dates. The have a gooey consistency that should give you a better mouth feel in your cookie.
I'm just making educated guesses here. But I hope they help.
Seeeeeeya; Goodweed of tthe North