Finally, we got the HX711 reading correctly. Still don't know whether the problem was with the HX711 or the scale/load cells we were using, but they refused to operate correctly at 3.3.v. We confirmed, from the Expressif official, that the ESP8266 is 5v input tolerant. The solution to our woes was to simply move the supply for the H X711 to +5v instead of 3.3v. Then everything worked fine, irrespective of whether we had modified the module for 3.3.v or not! At this time, in a controlled temperature, the scale is showing a 0.03% standard error (95.7 percentile). It was necessary to go to the raw Analog read function to ensure we were getting the correct data without any intervening HX711 adjustments.
When we did a least squares fit to various calibration weights, the equation was beautifully linear . Going forward, we will likely use these "raw A/D" numbers and do our own calibration adjustments just to avoid the unknown and ambiguity that the chip adjustments add to the calculations.
Based on these results it is now time to "polish" the software and begin final deployment. In the interim we may create a new calibration program though. It will be based on the raw A/D data.