This is of course, suggesting that the problem is related to the mismatch between the logic voltage and the operating voltage of the 74HC595s. The better way to approach this was to use the 74HCT14 to buffer between your ESP and the shift registers.
No, this is not sustainable as you are indeed "phantom powering" the shift registers and the ULNs which is way out of specification. You need to feed real 3.3 V to them from your ESP.
I still suspect however that the logic level mismatch is not the actual problem. 
And were you to rebuild, you should base it on TPIC6B595s.
Something we have not discussed so far, but just what has been powering the ESP all this time?