Don't some shortbread recipes use cornstarch? I don't know what proportion you could replace in an ordinary dough, but you could check out some shortbread recipes and follow their lead.
I think the only thing that using corn syrup would accomplish is making the cookies insipid. Honey and molasses have distinct flavors and corn syrup tastes like nothing, basically, except sweet.
What kind of cookies are you making that need such a fine texture? I'm intrigued.