  In Paper I of this series, we detected a significant value of the braking
index ($n$) for 19 young, high-$\dot{E}$ radio pulsars using $\sim$ 10 years of
timing observations from the 64-m Parkes radio telescope. Here we investigate
this result in more detail using a Bayesian pulsar timing framework to model
timing noise and to perform selection to distinguish between models containing
exponential glitch recovery and braking index signatures. We show that
consistent values of $n$ are maintained with the addition of substantial
archival data, even in the presence of glitches. We provide strong arguments
that our measurements are unlikely due to exponential recovery signals from
unseen glitches even though glitches play a key role in the evolution of a
pulsar's spin frequency. We conclude that, at least over decadal time scales,
the value of $n$ can be significantly larger than the canonical 3 and discuss
the implications for the evolution of pulsars.
