Monte Carlo computer-simulation techniques are used to elucidate the equilibrium phase behavior as well as the late-stage ordering dynamics of some two-dimensional models with ground-state ordering of a high degeneracy, Q. The models are Q-state Potts models with anisotropic grain-boundary potential on triangular lattices—essentially clock models, except that the potential is not a cosine, but a sine function of the angle between neighboring grain orientations. For not too small Q, these models display two thermally driven phase transitions, one which takes the system from a low-temperature Potts-ordered phase to an intermediate phase which lacks conventional long-range order, and another transition which takes the system to the high-temperature disordered phase. The linear nature of the sine potential used makes it a marginal case in the sense that it favors neither hard domain boundaries, like the standard Potts models do, nor a wetting of the boundaries, as the standard clock models do. Thermal fluctuations nevertheless cause wetting to occur for not too small temperatures. Specifically, we have studied models with Q=12 and 48. The models are quenched from infinite to zero as well as finite temperatures within the two low-temperature phases. The order parameter is a nonconserved quantity during these quenches. The nonequilibrium ordering process subsequent to the quench is studied as a function of time by calculating the interfacial energy, ΔE, associated with the entire grain-boundary network. The time evolution of this quantity is shown to obey the growth law, ΔE(t)∼t-n, over an extended time range at late times. It is found that the zero-temperature dynamics is characterized by a special exponent value which for the Q=48 model is n≃0.25 in accordance with earlier work. However, for quenches to finite temperatures in the Potts-ordered phase there is a distinct crossover to the classical Lifshitz-Allen-Cahn exponent value, n=1 / 2, for both values of Q. This supports the conjecture that the zero-temperature dynamics for models with soft domain boundaries belong to a special universality class, and that all models with nonconserved order parameter, independent of ordering degeneracy and softness and origin of domain boundaries, obey the classical growth law at finite temperatures. In quenches to the Potts-ordered phase vortices and antivortices occur and annihilate mutually without pinning the ordering process. The ordering dynamics for quenches into the intermediate phase is also found to be described by an effectively algebraic growth law.