First things first. Uninstall Prime95. Completely. Never use it again. I'll elaborate later.
The cooler is probably fine for what you're trying to do. I'd try lowering the voltage. 1.34V seems a tad on the high side for just 4.4GHz. If you're crashing then bad luck, but I think you can get lower. As Deders said, increase your LLC (load-line-calibration) and see what other power delivery options there are in your BIOS. Also disable any power-saving features as they can mess with your stability. Generally, you want everything set for better performance/overclocking, not low power consumption.
Other than that, general good practice when overclocking is to always set your voltage to manual when trying to determine a stable overclock. Really, you can't go wrong with that, as you remove the possibility of software messing with you from the equation. Also, set your RAM speeds to the default set by the CPU. If it's DDR4, that's 2133MHz. If it's DDR3, it's 1600MHz. Only put your RAM speeds back up (Assuming you have a faster kit) after finishing with the CPU overclock. Memory can cause weird issues that are really hard to diagnose. Been there, done that, never doing it again. While you're at it, you might want to test lowering your cache clock and voltage. That can sometimes help tremendoulsly with modern Intel CPU overclocking and the performance you lose is negligible.
Install OCCT and Asus RealBench. I've found these two to be the most useful stress testing tools by far. OCCT finds most kinds of instability without risking to kill your CPU and RealBench does the same but using a really heavy, but realistic load. You might also want to try out AIDA64, it's the most complete suite of testing and monitoring software out there. But let me tell you again, never use Prime95 again. It runs your CPU in a completely unrealistic scenario that uses every subsystem at the same time at 100%. The result is that modern Intel CPUs will throw more voltage in there to keep everything stable under that crazy AVX load. Some Haswell CPUs would even increase the core voltage by 0.1V or something crazy like that. Mine would do a 0.05V increase when I had already set it to 1.315V. And the craziest thing? It will still crash most of the time. So just get rid of it and use the other ones.
If you still have issues after doing all that... Then yeah, it's the IHS. I doubt it's THAT bad though.
Hope I helped. If you want more help, I'll be around.