Preheat oven to 350°F. Grease a 9x13-inch baking dish and set aside.
In a large bowl, combine crumbled cornbread and cubed white bread. Set aside.
In a large skillet, melt butter over medium heat. Add celery, onion, and bell pepper. Sauté until vegetables are soft, about 8-10 minutes.
Add garlic and cook for 1 minute more. Pour the vegetable mixture over the bread and toss to combine.
In a separate bowl, whisk together broth, eggs, sage, thyme, salt, pepper, and poultry seasoning.
Pour the broth mixture over the bread mixture and stir gently until everything is well moistened. The mixture should be wet but not soupy—add a bit more broth if needed.
Transfer the dressing to the prepared baking dish and spread evenly.
Cover with foil and bake for 30 minutes. Remove foil and bake for 20 more minutes until the top is golden brown and crispy. Let stand 10 minutes before serving.