You can test these things with a sensitive DMM - full scale output voltage is several mV given a few volts of excitation. Even 10μV resolution meter sees steps of 0.2% full scale or thereabouts.
No load offset voltage is usually specified as a very low percentage of full load output, this is easily verified.
My guess: you have permanently deformed it, resulting in change of offset voltage which the "bad" device somehow isn't able (or trying) to cope with.
Given your hypothesis, I figured at this point I might as well try squashing the aluminum load cell piece in a vise to see if that maybe counteracts any deformation incurred when I overloaded it with the hydraulic load.
IT DID SOMETHING!
I squashed it in a vise with some, but not a huge amount of force. I attached it back to the device, turned it on with no load, then put it in the reversed (tensioning) clamp to make a tensile load. I got up to 17kg tension before it overloaded and started beeping! Very encouraging.
The aluminum piece has 3mm wide slots cut into both sides in the middle. I went back to squash it even more in the vise, not sure how much force but at least 100-150kg if I had to guess, to the point where the slots reduced to ~2.41mm. I did it a few times, trying to center the piece so each slot showed equal reduction in width (if it was at all off center one slot showed more width than the other). I hesitated to apply more force than I did for fear of the aluminum plate with those slots cut in it, buckling up or down. Maybe I'll try again, sandwiching the aluminum piece in something so it can't buckle.
I hooked the reversed clamp back up and applied tension to the load cell again and got all the way up to 48kg. I think the clamp maxed out at that point since the load cell didn't start beeping to indicate an overload state.
I don't know if the reading is accurate anymore. I will try to find some known 10-20-30-40-50kg weights to test it with.
Even if it doesn't really work anymore, I'm happy to have an idea of where the problem lay. I am surprised such a tiny deformation of the aluminum, probably sub-mm, could cause such a problem.
Oh, also when I redid the grn-wht voltage check with no load it now reads 0.003V. Maybe I should take that reading between vise-squashing sessions to see if I can get it to 0? Would that indicate I had compensated all the deformation then?