I'm beginning to suspect the issue is in the card itself. Maybe the thermal paste didn't get properly applied or too little. Not likely, but possible.
Since the fans are speeding up in relation to the rising temp, I would not think the high fan speed was simply a BIOS error. Your case is huge and should not suffer from an air flow problem unless you live in extremely hot climes.
If replacing the thermal paste doesn't change anything, then I'd be inclined to believe you just got a hot chip.
If that's the case, maybe play around with the fan curve to have the fans ramp up sooner but to a lower, quieter RPM. Then go slowly up to 100% as temps increase. I have a reference R9-290 that I had to do that with. It was getting to 94C in almost every game and the fans noise was obnoxious. After playing with the fan curve, I now have it staying in the high 80Cs and much quieter.