Heat oven to 350 degrees. Butter a 9-by 13-inch baking dish.
Crumble cornbread into a large bowl. Tear toasted white or wheat bread into very small pieces, add to cornbread, and toss to combine
Melt 8 tablespoons butter in a large sauté pan over medium heat. Add onion, celery and bell pepper; sauté, stirring occasionally, for 3 minutes. Cover pan and cook, stirring occasionally, until vegetables are almost translucent, 5 to 6 minutes. Remove cover, add broth, and cook, scraping up any browned bits from the bottom of the pan, for 2 to 3 minutes. Continue to cook the mixture for 1 more minute, then remove from heat, add to bread mixture, and stir to combine.
In a medium bowl, stir together oysters, lemon juice, hot sauce, parsley, sage, salt and white pepper. Add to bread mixture and stir well to combine. If dressing seems too dry, add a little oyster liquor and up to ½ cup more chicken broth; mixture should be very moist.
Pour dressing into greased baking dish. Cut remaining 4 tablespoons butter into small pieces and scatter over top of dressing. Bake until top and sides are browned, 40 to 45 minutes.