  Understanding the orbits of spinning bodies in curved spacetime is important
for modeling binary black hole systems with small mass ratios. At zeroth order
in mass ratio, the smaller body moves on a geodesic. Post-geodesic effects are
needed to model the system accurately. One very important post-geodesic effect
is the gravitational self-force, which describes the small body's interaction
with its own contribution to a binary's spacetime. Another post-geodesic
effect, the spin-curvature force, is due to the smaller body's spin coupling to
spacetime curvature. In this paper, we combine the leading orbit-averaged
backreaction of point-particle gravitational-wave emission with the
spin-curvature force to construct the worldline and gravitational waveform for
a spinning body spiraling into a Kerr black hole. We use an osculating geodesic
integrator, which treats the worldline as evolution through a sequence of
geodesic orbits, as well as near-identity transformations, which eliminate
dependence on orbital phases, allowing for fast computation of inspirals. The
resulting inspirals and waveforms include all critical dynamical effects which
govern such systems (orbit and precession frequencies, inspiral, strong-field
gravitational-wave amplitudes), and as such form an effective first model for
the inspiral of spinning bodies into Kerr black holes. We emphasize that our
present calculation is not self consistent, since we neglect effects which
enter at the same order as effects we include. However, our analysis
demonstrates that the impact of spin-curvature forces can be incorporated into
EMRI waveform tools with relative ease. The calculation is sufficiently modular
that it should not be difficult to include neglected post-geodesic effects as
efficient tools for computing them become available. (Abridged)
