We study robust optimization problems with decision-dependent information discovery under polyhedral uncertainty and continuous recourse. We first establish structural properties of the resulting min–max–min problem, including existence of solutions and a condition under which information discovery has no value. Then, we develop a heuristic approach that combines affine decision rules with scenario generation and show that its adversarial separation problem is NP-hard. Additionally, we introduce a sampling procedure to estimate solution quality without solving the overall problem to global optimality. Numerical experiments on knapsack and facility-location instances show that the method outperforms a $K$-adaptability approach recently proposed in the literature in terms of computation time while achieving comparable solution quality. Our approach is applicable to polyhedral uncertainty sets that can be expressed as $\Xi = \Defset{ \sigma^+ – \sigma^- }{ (\sigma^+,\sigma^-)\in \Theta }$ with a downward monotone polytope $\Theta$.