Two-stage stochastic programming is a mathematical framework widely used in real-life applications such as power system operation planning, supply chains, logistics, inventory management, and financial planning. Since most of these problems cannot be solved analytically, decision-makers make use of numerical methods to obtain a near-optimal solution. Some applications rely on the implementation of non-converged and therefore sub-optimal solutions because of computational time or power limitations. In this context, the existing methods provide an optimistic solution whenever convergence is not attained. Optimistic solutions often generate high disappointment levels because they consistently underestimate the actual costs in the approximate objective function. To address this issue, we have developed two conservative-solution methodologies for two-stage stochastic linear programming problems with right-hand-side uncertainty and rectangular support: When the actual data-generating probability distribution is known, we propose a DRO problem based on partition-adapted conditional expectations whose complexity grows exponentially with the uncertainty dimensionality; When only historical observations of the uncertainty are available, we propose a DRO problem based on the Wasserstein metric to incorporate ambiguity over the actual data-generating probability distribution. For this latter approach, existing methods rely on dual vertex enumeration of the second-stage problem rendering the DRO problem intractable in practical applications. In this context, we propose algorithmic schemes to address the computational complexity of both approaches. Computational experiments are presented for the farmer problem, aircraft allocation problem, and the stochastic unit commitment problem.