They're usually pretty highly recommended as having good performance while being inexpensive. They're getting a little old now, but it would be a good upgrade over the stock cooler.
If the issue is a failing temperature sensor, a colder CPU should help keep it from shutting down, but I suspect you'll still have some issues under load.
If you're comfortable with it, a free and quick solution that would have a similar effect would be to lower CPU voltage to reduce temps. You would gain colder temperatures but it may make your processor unstable. If you accidentally turn voltage up, it could physically damage the processor or motherboard.
The process would be to find some kind of overclocking software (probably
AMD Overdrive which I've just noticed has been discontinued, so
here's a mirror) and lower the "CPU VID" voltage which will significantly reduce temperatures. You gain colder temps but the tradeoff is you reduce the stability of your processor. As long as you don't accidentally change your voltage up at all (shouldn't be over ~1.4v when you start out), there shouldn't be any risk.
You'll want to save and close out of anything you're working on and slowly lower CPU voltage down until either the temperatures look good or your computer crashes. If it does crash, when it boots back up, set voltage to slightly above where it crashed.
If that works, you could try both the new CPU cooler and lowered voltages and you should hopefully be back to pretty stable conditions.