We propose a novel approach to solve K-adaptability problems with convex objective and constraints and integer first-stage decisions. A logic-based Benders decomposition is applied to handle the first-stage decisions in a master problem, thus the sub-problem becomes a min-max-min robust combinatorial optimization problem that is solved via a double-oracle algorithm that iteratively generates adverse scenarios and recourse decisions and assigns scenarios to K subsets of the decisions by solving p-center problems. Extensions of the proposed approach to handle parameter uncertainty in both the first-stage objective and the second-stage constraints are also provided. We show that the proposed algorithm converges to an optimal solution and terminates in finite number of iterations. Numerical results obtained from experiments on benchmark instances of the adaptive shortest path problem, the regular knapsack problem, and a generic K-adaptability problem demonstrate the performance advantage of the proposed approach when compared to state-of-the-art methods in the literature.