Its clearly the new fan since its not connected to the GPU. Therefore the GPU can't regulate the fan speed and it crashes. Its simple as that. you didn't do due diligence before buying a new cooling solution and now you are paying the price.
If you have the old cooler(assuming it works and you replaced it just to have better cooling) then put that back in.
I'm assuming the connector on the GPU is with 3 pins and the new cooler has 4 pins? If its the other way around you could still plug the 3 pins in the 4 pins on the GPU, just make sure the right side end is placed to the most right side of the connector on the GPU as well, usually the last pin is the sensors, while the others are for the power for the fans.
Also are you checking the temperature during playing games? Idle temperature is always going to be low no matter what, you could literally have no cooler and idle temperature would still be low, so you need to check it when the GPU is being fully used in games!
To give some more depth here, and some new information I was able to pull with another utility, the GPU with the old cooler was idle around 56 degrees and got far too hot during play. Artifacting started showing up, so I stopped using it until I got a new cooling solution so I wouldn't continue likely damaging the card.
When running a game with it at a somewhat low load it goes from idling between 30 and 40 to about 52 so far, but I don't know how much higher it will go since it crashes out.
I'm glad it's likely the fan connection, I have an adapter on the way to go from the mini GPU 4 pin to the TX3 on the new fan.
A proper adapter should as far as I know bridge the sensing pin so it just sees the fan as always at 100%.
The new data shows that the card is getting a very stable 12.16 volts, fluctuating down as low as 12.16 and as high as 12.18, right up until it shuts off. VRM temperatures are low, never exceeding 43 degrees so far.