After about a week of waiting, the Adafruit DRV8871 H-bridge motor driver breakouts were delivered. They will replace the MOSFET circuit driving the pumps, which were allowing all kinds of noise and voltage spikes and caused the TPIC shift registers to reset every instance the pumps were switched on. Additionally, the previous electronics setup was mysteriously allowing the valves to leak water through to the sample bags when switched off. With the pumps being driven by the H-bridge, this problem has completely disappeared. The original cause was most likely related to the valves "soft resetting", where each valve was neither completely off nor completely on after the pump was powered.
We just received an email today letting us know that the rest of the valves have finally shipped, which will allow us to completely assemble our 24 bag sampler! While we wait for that, I'll be working on translating the electronics to a PCB and getting that printed. I'll also be reviewing the design and adding more functionality to the code.
I recorded of a video of the system in action, included below, which is the first recorded successful test of our water sampler!