Preheat oven to 300℉.
Cut the cornbread and white bread into 3/4 inch cubes. (I find that the cornbread bakes up thick, so I slice the entire loaf into cubes, then lay each row over and slice in half lengthwise again to get just the right size)
Spread onto a large baking sheet in a single layer and bake at 300 for 30-60 minutes (usually takes about 45 mins) until dried out, gently shuffle the bread cubes about 20 minutes through baking. (Bread should be fairly dried out and crisp to the touch, like a crouton. If not dried out the dressing will be mushy. The cornbread will not be as crisp as the white bread, but should still be fairly dried)
Preheat oven to 375℉.
In a medium size saucepan melt the butter over medium heat. Add the onion and celery and cook until softened.
Stir in the herbs, salt and pepper and pour in the chicken broth. Heat until mixture comes to a simmer. Taste and adjust seasonings/herbs as needed.
Add the bread crumbs to a 2 1/2 qt baking dish, ladle broth mixture over the bread crumbs until entire mixture is moistened, not soggy.
Beat the eggs, pour over the dressing, gently stir the dressing to coat the bread crumbs (be gentle since the cornbread will easily crumble)
Cover with foil and Bake for 20-25 at 375, remove foil and bake an additional 15 mins.