We consider dual subgradient methods for solving (nonsmooth) convex constrained optimization problems. Our focus is on generating approximate primal solutions with performance guarantees and providing convergence rate analysis. We propose and analyze methods that use averaging schemes to generate approximate primal optimal solutions. We provide estimates on the convergence rate of the generated primal solutions in terms of both the amount of feasibility violation and bounds on the primal function values. The feasibility violation and primal value estimates are given per iteration, thus providing practical stopping criteria. We provide a numerical example that illustrate the performance of the subgradient methods with averaging in a network resource allocation problem.