Low thermal conductive semiconductors have attracted huge attention for heat management and heat harvesting applications. Although the weak chemical bonding in the Cu/Ag-based chalcogenides is promising in suppressing heat transport, their quaternary analogs remain less explored. Here, we report on a comparative study of the crystal structure and thermal conductivity of various Ag-containing variants of Cu₄TiSe₄, i.e., Ag_1-xCu_3+xTiSe₄ (x = 0−0.8) samples. Analysis of the crystal structure, phase transition, and temperature-dependent lattice thermal conductivity (κL) of pristine AgCu₃TiSe₄ and Ag_1-xCu_3+xTiSe₄ have been carried out both experimentally and theoretically. The cubic crystal structure (space group P4̅3m) of these Ag variants is identical to that of Cu₄TiSe₄ or Cu₄TiTe₄, where a positionally disordered Cu sublattice is either replaced by Ag (for AgCu₃TiSe₄) or by Ag/Cu substructure (for Ag_1-xCu_3+xTiSe₄). Upon cooling, the symmetry reduction to a rhombohedral (space group R3m) structure is attributed to the partial ordering of the positionally disordered Ag. The proposed structural models at different temperatures have been further analyzed using the Maximum Entropy Method (MEM). X-ray photoelectron spectroscopy measurement suggests that the parent compound forms a charge-precise (Ag⁺)(Cu⁺)₃(Ti⁴⁺)(Se²⁻)₄ chemical formula. Interestingly, the lattice thermal conductivity of the Ag_1-xCu_3+xTiSe₄ samples remains very low, with values varying in the range of ∼0.65−0.24 W m⁻¹ K⁻¹ between 293 and 623 K. Density Functional Theory (DFT) calculation shows the presence of antibonding states of Cu(3d)/Ag(4d)−Se(4p) below the Fermi level (E_F), providing softness to the lattice of AgCu₃TiSe₄. In addition, the positional disordered site plays a crucial role in further softening the framework and provides large lattice anharmonicity. The calculated phonon dispersions evidence the presence of several soft optical phonon modes at ca. 25 cm⁻¹, originating from the atomic vibrations of Ag, Cu, and Se. Further confirmation of these phonon modes is obtained from the low-temperature heat capacity study. The low-lying optical phonon modes in AgCu₃TiSe₄ are mainly caused by the presence of a soft lattice framework, positional disorder and associated rattling-like vibrations of Ag⁺/Cu⁺ ions. Their strong interaction with the heat-carrying acoustic phonon modes is key ingredient that explains the very low κ_L.