Your understanding is correct. The doppler shift observed from a galaxy is the sum of its peculiar velocity with respect to the "Hubble flow" and the redshift due to the Hubble flow, which is caused by the expansion of the universe.
There is no direct way from a spectrum to separate these two components - they have the same qualitative result.
In principle, the expansion of the universe (or a change in the peculiar velocity) could be directly measured by looking for a change in redshift with time, which would depend on the cosmological parameters.
This is an extremely small effect and is confused by the peculiar motions of individual galaxies. Nevertheless, measuring this redshift drift is one of the prime goals of the Codex Instrument on the E-ELT (see Pasquini et al. 2010, http://esoads.eso.org/abs/2010Msngr.140...20P )
using Lyman alpha absorption systems towards distant quasars. This experiment is also planned for the Square Kilometre Array, using the 21cm line (Kloeckner et al. 2015 http://arxiv.org/abs/1501.03822 ).
In both cases, to overcome the experimental uncertainties (eg at 21 cm, it amounts to line drifts of 0.1 Hz over a decade), then observations of millions of galaxies must be combined.
There is no prospect of measuring this effect in an individual galaxy, furthermore I fear your understanding of cosmological redshift is flawed. The dependence on distance is a statistical average, not an absolute dependence. Individual galaxies are moving in individual gravitational potentials from objects around them. This gives them their peculiar velocities with respect to the flow. This velocity could increase or decrease as a galaxy got further away, but is never expected to be large enough to be detectable on human timescales for any individual galaxy. In addition, any change in peculiar velocity should average to zero when looking at millions of galaxies, leaving the redshift drift due to the expansion.
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