We study the non-equilibrium dynamics of the spin-1/2 anisotropic Heisenberg (XXZ) spin chain starting from Majumdar-Ghosh dimer product and N\'eel states. Using the quench action formalism based on the exact overlaps between the initial state and the eigenstates of the Hamiltonian, we determine the string distribution functions characterizing the asymptotic state and compute various short range correlation functions in the large time limit. Our results are in excellent agreement with matrix product state based real time simulations of the time evolution. In contrast, the Generalized Gibbs Ensemble (GGE) predictions significantly deviate from the numerics for the dimer initial state. These findings suggest that while the quench action formalism correctly captures the steady state, the GGE cannot give a complete description even of local observables.
