A core temp of 50c is not likely the reason that your PC is shutting down. Something other than the CPU is likely the issue. When the FX-8xxx CPUs reach 70C they automatically throttle the CPU vcore voltage and frequency to prevent higher temps. The FX-8350 CPU would not shut the system down until ~90C. AMD's official 24/7 core temp for the FX-8350 is 61.1C and I have tested them to 70+ C without system shut down.
I'm guessing that your mobo, PSU, RAM or video card might be acting up with the increased load from the FX-8350. You can try running Memtest86+ V4.20 or V5.0 on the DIMMs overnight to see if they pass. You can also test with just one DIMM if you have multiple DIMMs. You can even reduce the RAM frequency if it's above 1600 MHz.
Make sure that your PSU can handle the FX-8350 which draws a lot of power under heavy usage. Also check to see what the GPU temp looks like full load. Re-seating the DIMMS/GPU card/CPU and cables is a good first step. Manually setting the RAM timings, voltages, LLC, etc. in BIOS may also be necessary with the FX-8xxx series CPUs for complete system stability.
If the mobo VRM is overheating, which it could be after 30 minutes of video game or stress testing, it can cause your PC to shut down. You can test this by adding a fan that blows cool air on the VRM heatsink.
BTW, while the H80i is not a great cooler it should be able to handle the FX-8350 at the stock frequency.
Building a reliable PC involves more than just assembling the parts. You need to be able to configure all of the BIOS settings appropriately. This can be quite involved and frustrating as it can require a lot of trial and error with stress testing. It is however often the only means to get a 100% reliable PC.