The primary purpose of this paper is to introduce the effect of chain stretching into a previously developed, thermodynamically admissible reptation model incorporating anisotropic tube cross sections, double reptation, and convective constraint release, while avoiding the independent alignment approximation. A second goal is the detailed illustration of the thermodynamic modeling approach. Two versions of the model with different stretching mechanisms are proposed, and the simpler one sheds new light on thermodynamically admissible reptation models without independent alignment. The stochastic reformulation of the new model, its simulation, its linear viscoelastic properties, its predictions for rapid double-step shear strains, and the model parameters are discussed in detail.