Osd is on screen display. It's an overlay that sits in the top left of the screen while you are actually in-game and does real-time temps for cores gpu etc, however you set it. It's also far more accurate than SpeedFan usually is as far as actual temps go. Really ideal for ppl with a single monitor. Running dual monitors gets me multiple other options as I'm not reliant on max temps.
You are getting shutdowns. That generally doesn't happen unless the cpu has already throttled, a bunch, and there's no hope in sight. Only then does the cpu call it quits. There's usually 2 inherent causes for this, lack of sufficient cooler size and lack of airflow. You can have a great cooler like a Noctua NH-D15S and still get temps skyrocketing if there's insufficient airflow and the pc case has become an oven. If it's 70° inside the case, the cooler can't cool the cpu below 70°. So case fans and decent airflow can play a large part in keeping your pc cool under loads.
OC is not necessarily a bad thing. Your cpu currently is running on stock voltages. Those voltages are intentionally set overly high by Intel, to guarantee stability of every 2500 cpu. Each cpu is different, and responds differently to different voltages, all part of the silicon lottery. And it covers turbo too.
My i5-3570k stock voltages are @ 1.25v. Base clock is 3.4GHz, turbo is 3.8GHz. Same voltage. I can set the multiplier to 4.2GHz and still get the same voltages and temps. However, if I manually change the vcore from auto to 1.114v my temps go from idle/game/stress of 32/60/80 to 32/55/70. And that's at 4.3GHz, 500MHz above turbo with lower than stock auto voltages.
So just saying you have no OC, or assuming that OC will raise temps is not quite right, for a small-mild OC you can actually lower temps overall.
I already gave some other cooler options, the TX3 will work, but it's on the low end of the scale, it's a 10 year old design using a fan that was ok 10 years ago, but there are newer designs that are more technologically advanced for better performance. The Cryorig m9i being a perfect example, it competes at the same level as the 120mm hyper212, the 90mm TX3 big brother. And beats it. As does the Raijintek Aidos, both 92mm coolers. With an i7-4770k at stock and OC.
https://www.techpowerup.com/reviews/CRYORIG/M9i/6.html
The TX3 runs @ 5°+ hotter than the 212.
Can it do the job, sure, its going to be a good 10 - 15° cooler than Intel stock, but if you are coming close to 100° and shutdowns, 85-90° isn't all that much better.