Yes, if you bought a new Desktop Base, everything should work perfectly fine, if you have the money to buy a new one.
You're pretty much buying a new computer in general.
If you have the money, I reccommend something with these system-specs...
1 or 2 GB (Gigabytes) of RAM
Windows 7
Anything with 100 or more Gigabytes of Hard Drive Memory is good. Everything has 320 GB nowadays.
So if you can at least get those things, you're set. If you want to play games that were made from 2008, last year, or even this year, you're going to want something with a thing called NVIDIA or ATI RADEON. These two things are for computer graphics needed to play games.
Cheap and Good laptops are hard to come by these days, especially if you're looking for Windows XP.
From what you've told me, if you're computer is turning off on it's own, it's probably a Computer Virus.
You can either:
1. Find an anti-virus program. (If you pay money for anti-virus, it's better because it actually works)
Free anti virus is available, it's just those programs are not all that great.
or 2. Format your Hard Drive and reinstall the computer system altogether. And by the looks of your question, you're not really good with computers huh? So if you do that, you are going to have to learn a lot on how to do that.
Other than that, try uninstalling programs by going to 'Control Panel' and 'Add Remove Programs'
and get rid of what you don't want.
You can try to put your computer in front of an air conditioner or something like a fan.
If the problem truely is Overheating, you need to keep your room cool.
Overheating might also be because of your computer's age. If you have an old Windows 95, 98, or 2000, I can see overheating being a problem, but if it isn't and your room is cool, this isn't the problem.
That's all the help I can give.