Fill a large pot with water. Add 3 tablespoons of the lime juice and 2 tablespoons salt and bring to a boil.
Add the shrimp and cook until opaque throughout, 2 to 3 minutes. Transfer the shrimp to a bowl of ice water to cool. Drain and pat dry; set aside.
In a large bowl, combine the sour cream, cayenne, remaining tablespoon lime juice, and ½ teaspoon salt.
Add the carrots, apple, and cilantro and toss to combine. Serve with the shrimp.