Recent high-resolution observations of interstellar absorption lines of CH and CN toward Zeta Oph obtained by Crane et al. (1986), and Palazzi et al. (1988), exhibit line widths that suggest thermal line broadening at high temperature, T about 1200 K. Observations of CO line emission at 2.6 mm toward Zeta Oph (Langer et al.,1987) indicate that the molecular gas resides in four distinct velocity components that span less than 3 km/s in Doppler velocity. Simulated CH and CN absorption line profiles are compared for high-temperature (T = 1200 K) thermal broadening and for a combination of low-temperature (T = 50 K) thermal plus turbulent broadening. It is shown that the two broadening models reproduce existing observations comparably well and are virtually indistinguishable at a lambda/Delta-lambda ratio of about 100,000. The observed differences in the CH and CN line widths may reflect slightly different distributions of those molecules along the line of sight. The simulations use very recent, improved laboratory spectroscopic data on CH (Bernath). Some related consequences of such unresolved velocity structure on the ultraviolet absorption lines of CO are examined. Indirect diagnostics of temperature in the Zeta Oph cloud favor low-temperature thermal plus turbulent broadening, and the implied rate of dissipation of turbulence is in harmony with estimates of the global input of mechanical energy into to interstellar medium.