In the general scheme of things, ignoring a failure within the processor or within the DIMMs themselves (though these are both possibilities for point of failure as well), the memory channels can be disaffected by noise introduced by the processor, noise introduced by the motherboard and noise introduced by the DIMMs. In many cases, the reason why folks have more problems when using 4 DIMMs (instead of two) is because of noise (the additional noise from having a second DIMM on a channel pushes things past the noise failure thresholds). Secondly, putting a single DIMM in the black connector can be problematic because of reflections occurring behind the DIMM (at the blue connector) (and yes, trying this was a desperation suggestion). Thirdly, as a motherboard ages, noise will increase on the memory channels due to oxidation, component degradation, etc. and etc. Your board is approaching 6 board generations out of date and 7 years of age - which is the typical MTBF (Mean-Time Between Failures) for most of the components used.
Bottom line, you likely are having intermittent errors occurring on the memory channel(s). You could try using a different set of DIMMs, in order to see whether this improves the noise situation (and, as I said, it could still be a failing DIMM issue), but my recommendation is to not throw good money after bad. It is time to consider replacing this equipment completely...
Hope this helps,
...S