  Excited state contamination remains one of the most challenging sources of
systematic uncertainty to control in lattice QCD calculations of nucleon matrix
elements and form factors. Most lattice QCD collaborations advocate for the use
of high-statistics calculations at large time separations ($t_{\rm
sep}\gtrsim1$ fm) to combat the signal-to-noise degradation. In this work we
demonstrate that, for the nucleon axial charge, $g_A$, the alternative strategy
of utilizing a large number of relatively low-statistics calculations at short
to medium time separations ($0.2\lesssim t_{\rm sep}\lesssim1$ fm), combined
with a multi-state analysis, provides a more robust and economical method of
quantifying and controlling the excited state systematic uncertainty, including
correlated late-time fluctuations that may bias the ground state. We show that
two classes of excited states largely cancel in the ratio of the three-point to
two-point functions, leaving the third class, the transition matrix elements,
as the dominant source of contamination. On an $m_\pi\approx310$ MeV ensemble,
we observe the expected exponential suppression of excited state contamination
in the Feynman-Hellmann correlation function relative to the standard
three-point function; the excited states of the regular three-point function
reduce to the 1% level for $t_{\rm sep} >2$ fm while, for the Feynman-Hellmann
correlation function, they are suppressed to 1% at $t_{\rm sep}\approx1$ fm.
Independent analyses of the three-point and Feynman-Hellmann correlators yield
consistent results for the ground state. However, a combined analysis allows
for a more detailed and robust understanding of the excited state
contamination, improving the demonstration that the ground state parameters are
stable against variations in the excited state model, the number of excited
states, and the truncation of early-time or late-time numerical data.
