You refrigerate oatmeal cookies is because cold dough spreads more slowly in the oven so the cookie is thicker and more moist. In this case, it has nothing to do with the thickness of the dough.
I love these. One of the better oatmeal cookies I've tried, and I'm an avid baker! That holiday 2001 edition magazine is pure gold. It is my go to recipe source for cookies, year-round.
First, check your instructions. You mention MILK(line 4) and there is no milk in the recipe. I searched the internet to confirm. It is confusing. The cookie itself is bland. I believe it needs more spices. It also didn't need any refrigeration prior to baking. The dough is very stiff. I used my hands for the last step. I think there are better recipes out there for oatmeal cookies. I don't know how other folks feel, but I have never had real success with Martha's recipes with one exception, her gingerbread. Thanks!!