We report temperature-dependent transport and x-ray diffraction measurements of the influence of Ti hole doping on the charge density wave (CDW) in $1T\text{\ensuremath{-}}{\mathrm{Ta}}_{1\ensuremath{-}x}{\mathrm{Ti}}_{x}{\mathrm{S}}_{2}$. Confirming past studies, we find that even trace impurities eliminate the low-temperature commensurate (C) phase in this system. Surprisingly, the magnitude of the in-plane component of the CDW wave vector in the nearly commensurate (NC) phase does not change significantly with Ti concentration, as might be expected from a changing Fermi surface volume. Instead, the angle of the CDW in the basal plane rotates, from $11.9{}^{\ensuremath{\circ}}$ at $x=0$ to $16.4{}^{\ensuremath{\circ}}$ at $x=0.12$. Ti substitution also leads to an extended region of coexistence between incommensurate (IC) and NC phases, indicating heterogeneous nucleation near the transition. Finally, we explain a resistive anomaly originally observed by Di Salvo [F. J. Di Salvo et al., Phys. Rev. B 12, 2220 (1975)] as arising from pinning of the CDW on the crystal lattice. Our study highlights the importance of commensuration effects in the NC phase, particularly at $x\ensuremath{\sim}0.08$.