  We study the spin dynamics of diamond nitrogen vacancy (NV) centers in an
oscillating magnetic field along the symmetry axis of the NV in the presence of
transverse magnetic fields. It is well-known that the coupling between the
otherwise degenerate Zeeman levels $|M_S=\pm1\rangle$ due to strain and
electric fields is responsible for a Landau-Zener process near the
pseudo-crossing of the adiabatic energy levels when the axial component of the
oscillating magnetic field changes sign. We derive an effective two-level
Hamiltonian for the NV system that includes coupling between the two levels via
virtual transitions into the third far-detuned level $|M_S=0\rangle$ induced by
transverse magnetic fields. This coupling adds to the coupling due to strain
and electric fields, with a phase that depends on the direction of the
transverse field in the plane perpendicular to the NV axis. Hence, the {\em
total coupling} of the Zeeman levels can be tuned to control the adiabaticity
of spin dynamics by fully or partially compensating the effect of the strain
and electric fields, or by enhancing it. Moreover, by varying the strength and
direction of the transverse magnetic fields, one can determine the strength and
direction of the local strain and electric fields at the position of the NV
center, and even the {\em external} stress and electric field. The nuclear spin
hyperfine interaction is shown to introduce a nuclear spin dependent offset of
the axial magnetic field for which the pseudo-crossing occurs, while the
adiabaticity remains unaffected by the nuclear spin. If the NV center is
coupled to the environment, modeled by a bath with a Gaussian white noise
spectrum, as appropriate for NVs near the diamond surface, then the spin
dynamics is accompanied by relaxation of the Zeeman level populations and
decoherence with a non-monotonic decrease of the purity of the system.
