The cubic semiconducting compounds APd(3)O(4) (A = Ca, Sr) can be hole-doped by Na substitution on the A site and driven toward more conducting states. This process has been followed here by a number of experimental techniques to understand the evolution of electronic properties. While an insulator-to-metal transition is observed in Ca1-xNaxPd3O4 for x >= 0.15, bulk metallic behavior is not observed for Sr1-xNaxPd3O4 up to x = 0.20. Given the very similar crystal and (calculated) electronic structures of the two materials, the distinct behavior is a matter of interest. We present evidence of local disorder in the A = Sr materials through the analysis of the neutron pair distribution function, which is potentially at the heart of the distinct behavior. Solid-state Na-23 nuclear magnetic resonance studies additionally suggest a percolative insulator-to-metal transition mechanism, wherein presumably small regions with a signal resembling metallic NaPd3O4 form almost immediately upon Na substitution, and this signal grows monotonically with substitution. Some signatures of increased local disorder and a propensity for Na clustering are seen in the A = Sr compounds.