  We continue our investigations of the development and importance of the
one-arm spiral instability in long-lived hypermassive neutron stars (HMNSs)
formed in dynamical capture binary neutron star mergers. Employing hydrodynamic
simulations in full general relativity, we find that the one-arm instability is
generic in that it can develop in HMNSs within a few tens of milliseconds after
merger for all equations of state in our survey. We find that mergers with
stiffer equations of state tend to produce HMNSs with stronger $m=2$ azimuthal
mode density deformations, and weaker $m=1$ components, relative to softer
equations of state. We also find that for equations of state that can give rise
to double-core HMNSs, large $m=1$ density modes can already be present due to
asymmetries in the two cores. This results in the generation of $l=2$, $m=1$
gravitational wave modes even before the dominance of a one-arm mode that
ultimately arises following merger of the two cores. Our results further
suggest that stiffer equations of state give rise to HMNSs generating lower
$m=1$ gravitational wave frequencies. Thus, if gravitational waves from the
one-arm instability are detected, they could in principle constrain the neutron
star equation of state. We estimate that, depending on the equation of state,
the one-arm mode could potentially be detectable by second generation
gravitational wave detectors at $\sim 10$ Mpc and by third generation ones at
$\sim 100$ Mpc. Finally, we provide estimates of the properties of dynamical
ejecta, as well as the accompanying kilonovae signatures.
