In a 10- to 12-inch frying pan over high heat, combine butter, onions, celery, and corn. Stir often until vegetables are lightly browned, 5 to 8 minutes. Pour into a large bowl.
Crumble sausage in frying pan and stir often until lightly browned, about 5 minutes. With a slotted spoon, transfer sausage to onion mixture. Discard fat.
Pour oysters and liquid into frying pan. Stir to free browned bits and bring to a boil. With slotted spoon, lift out oysters and cut into 1/2-inch chunks. Add to onion mixture. Boil oyster liquid until reduced to 2 tablespoons and add to onion mixture. Add sage, thyme, and cornbread; mix well, breaking cornbread into small pieces. Season to taste with salt and pepper.
Pat into a shallow 3-quart (9- by 13-in.) casserole. For moist dressing, cover with foil; for crusty dressing, do not cover. Bake in a 325° to 350° oven until hot (at least 150° in center) or lightly browned, about 50 minutes (1 hour if chilled).