Temps in the 60+ (C) on Prime 95 or heavy gaming don't sound surprising with a stock CPU cooler. I don't think they would be damaging the CPU, either.
There are several things that you could do, depending on your case, your motheboard, etc.
One thing that you could do on many motherboards is go into the fan profile in BIOS and create a manual profile that runs the CPU fan at 100% speed at 60C. My ASUS default setting didn't crank up the fan to full speed until the CPU hit 70C. With a stock cooler, you don't really want to give up cooling power like that. The stock cooler relies on high fan speeds to keep the CPU cool.
For CPU intensive high-heat operation, an aftermarket CPU cooler will perform better and/or quieter than the stock cooler, so that's a possible upgrade to consider.