We develop an exact solution framework for a broad class of Distributionally Robust Optimization (DRO) problems with general uncertainty structure. Within the class of moment- and confidence-set-based ambiguity sets, existing exact methods are largely limited to max-of-affine functions under ambiguity sets with strictly nested confidence sets. To enlarge this scope while preserving tractability, we introduce an alternative ambiguity set based on linearly defined confidence sets that allows for weak nestedness. We then consider DRO problems whose functions are convex in the decision variables and satisfy one of the following three cases with respect to the uncertainty: (i) when the function is convex under the nested general ambiguity set, we develop a global optimization algorithm; (ii) when the function is generally nonlinear, we show that the same algorithm applies under the alternative ambiguity set; and (iii) when the function is concave, we derive an explicit conic reformulation under this alternative ambiguity set. All three cases are handled by first reformulating the DRO problem as a robust problem and then applying advanced techniques from robust optimization. By solving these problems to optimality, our framework can offer valuable insights into the conservatism and behavior of existing approximation-based DRO models. We illustrate the generality and practical relevance of the proposed framework through two applications in capital budgeting and appointment scheduling.