  We reexamine $k$-essence dark energy models with a scalar field $\phi$ and a
factorized Lagrangian, $\mathcal L = V(\phi)F(X)$, with $X = \frac{1}{2}
\nabla_\mu \phi \nabla^\mu \phi.$ A value of the equation of state parameter,
$w$, near $-1$ requires either $X \approx 0$ or $dF/dX \approx 0$. Previous
work showed that thawing models with $X \approx 0$ evolve along a set of unique
trajectories for $w(a)$, while those with $dF/dX \approx 0$ can result in a
variety of different forms for $w(a)$. We show that if $dV/d\phi$ is small and
$(1/V)(dV/d\phi)$ is roughly constant, then the latter models also converge
toward a single unique set of behaviors for $w(a)$, different from those with
$X \approx 0$. We derive the functional form for $w(a)$ in this case, determine
the conditions on $V(\phi)$ for which it applies, and present observational
constraints on this new class of models. We note that $k$-essence models with
$dF/dX \approx 0$ correspond to a dark energy sound speed $c_s^2 \approx 0$.
