I know these are basic and troublesome, but:
- have you tried re-applying thermal compound?
- have you dusted out your computer? This is a recent problem as you've said, might be the case
Before this recent problem, what kind of temperatures are you seeing? I would get a better cooler later for your CPU because it deserves it and you certainly don't want it to always be hot during gaming. Have you capped the frame-rate to your monitor's refresh rate (I assume 60hz)?
It's not like it's an absolute need, seeing as how you haven't mentioned it going past 85, but please do try the things I've mentioned. There are plenty of good (within reason) coolers that would meet a budget and expectations.
(To answer the question, it is safe to...