Preheat the oven to 375 degrees F. Butter a 3-quart baking dish. Melt 6 tablespoons butter in a large skillet over medium heat; add the ham and cook, stirring, until lightly browned, about 4 minutes. Add the onion, celery, bell pepper, tarragon, thyme and Old Bay and season with salt and pepper; cook, stirring, until the vegetables are tender, about 5 minutes. Add the broth and bring to a simmer.
Whisk the eggs and parsley in a large bowl. Add the cornbread, white bread and shrimp, then pour in the vegetable-broth mixture and gently toss to combine.
Transfer the stuffing to the prepared baking dish and dot with the remaining 1 tablespoon butter. Cover with aluminum foil and bake 30 minutes, then uncover and bake until golden, about 30 more minutes.