If you want soft and moist, you need egg yolks. Their emulsifier and fat content makes dough pliable, soft and smooth, and retains moisture.
Egg whites dry out a dough. This is sometimes desirable, e.g. in pate a choux. Eclairs made with whole eggs often have wet planes in the middle, resulting in an underbaked impression. If you remove some yolks from the dough and use a mixture of whole eggs and egg whites, you get a firmer, drier dough. But in many cases, people want softer, moister end products, and in such cases, recipes which increase the ratio of egg yolks to egg whites (by using more yolks than whole eggs, or yolks only) will give you a better result.
Egg yolks and egg whites have very different roles in baking. You can almost never replace them. And in the cases in which you can, you will end up with a different texture.
In this case, it won't be sure disaster to replace, but I would be very reluctant to do it. Flourless recipes are finicky. Flour holds stuff together. Nut flours don't hold anything together, they need a binding agent. Egg whites are a good binding agent. Egg yolks are not only not good as a binding agent (except in certain circumstances, for example in custards - but you don't have this here), they are even a lubricant and as such will interfere with the binding done by the egg whites.
If you substitute here, you will not only change the taste, you risk your cookies crumbling apart in your hands.
In general, don't replace stuff in baking recipes. They are hard to design right. Even if you have some issues (e.g. allergies), it is easier to find an existing recipe without the offending ingredient than to try to tweak an existing one to work with different ingredients. Eggs are especially hard to substitute. Such a reason like "not wanting to waste" loses its sense if what you end up with is bad cookies, which are a much worse waste of products. And besides, you can do many more interesting things with a yolk than just scrambled eggs. In fact, I am frequently throwing out whites because I have found no use for them after doing something with the yolks.
If you still want to go ahead and try if the cookies work for you, try using 3 whole eggs. 2 eggs are nowhere near enough to substitute for 4 egg whites. Not only are the yolks smaller than the whites volume-wise, they will also not deliver you the binding proteins you need so badly.
Best Answer
In principle, you'd probably be able to find a recipe that uses ~2 eggs with a similar net volume, but it may not be so easy to retrofit your existing recipe, as it's reasonable to expect that the recipe is relying on some property of the yolk (emulsifying power, fat content, etc). You may find the result satisfying, but it is likely that you will produce a different texture than the recipe intended.
Given my experience with cake baking, I can say it's likely easier to substitute whole eggs for yolks than, for example, trying to use whole eggs in place of whites. Egg whites foamed into a meringue structure can be used to provide structure to a cake in place of leavening, for example, but whole eggs will not replicate that structure without being separated.
On the other hand, since an egg yolk does not generally provide structure as much as it does flavor, other than the extent which emulsifying power constitutes structure, it's fairly likely that you can get a pleasing cake that will perhaps be less dense than the egg yolk-only version.
Keep in mind that the egg white constitutes approximately 2/3 of the weight of an egg; the density should be similar, however. So it's quite possible you'll need only 4/3 eggs to provide similar volume of egg product; it's up to you whether you'd want to split the egg, round up to 2 eggs, or use one egg plus one yolk.
I should add, though, that there are plenty of ways to use up surplus egg whites if you make the recipe as written:
You can freeze egg whites in ice cube trays if, like me, your best intentions to use up the surplus are likely to be delayed for whatever reason.