  Non-diagonal couplings to a bosonic bath completely change polaronic
dynamics, from the usual diagonally-coupled paradigm of smoothly-varying
properties. We study, using analytic and numerical methods, a model having both
diagonal Holstein and non-diagonal Su-Schrieffer-Heeger (SSH) couplings. The
critical coupling found previously in the pure SSH model, at which the $k=0$
effective mass diverges, now becomes a transition line in the coupling constant
plane - the form of the line depends on the adiabaticity parameter. Detailed
results are given for the quasiparticle and ground state properties, over a
wide range of couplings and adiabaticity ratios. The new paradigm involves a
destabilization, at the transition line, of the simple Holstein polaron to one
with a finite ground-state momentum, but with everywhere a continuously
evolving band shape. No 'self-trapping transition' exists in any of these
models. The physics may be understood entirely in terms of competition between
different hopping terms in a simple renormalized effective band theory. The
possibility of further transitions is suggested by the results.
