After playing in Live Mode for about 5 minutes, no matter how many Sims or what the size of my lot is, all of my Sim's clothing, hair, and skin will suddenly get horribly blurry.
I have all my graphics set to High, and still the textures keep getting worse and worse. I have attached some images so you can see what I'm talking about.

I know it's a bit hard to see, but notice how their lips and eyes look blurry, and how the colors are spilling out of their clothing a bit.
In case you think I'm overreacting, it gets worse than what's in the pictures.

Look at how nice and clear this Sim's face is: Picture
My game only stays that way for 5 minutes.

I have looked at all other threads I could find about this topic. I have cleaned out most of the dust in my computer, and tried doing what this thread said. (Although, my computer has an ATI Graphics Card, not an Nvdia like the thread was meant for.)

My computer has an ATI Radeon 4350.
So, anyone have any advice for what I should do about this problem?

Your card can not handle setting on high for it is a very low end video card and right at the very low end of specs.
You will either need to set your setting on low or get a better video card.

It starts of clear and then because you are on a high graphic setting which it can't handle it gets hot. You could be doing damage to your computer. Also the more cc the more it has to work as well. Things to try are turning shadows, reflections, neighbourhood deco and neighbours off. After that you need to start turning your graphics down. But really you need a better card.

I find this kind of strange because the game didn't start doing this right up until about 2 months ago. Aside from the blurring the game runs perfectly with no lag at all, so I wouldn't have ever guessed it'd be my card...

Up until 2 months ago, you hadn't managed to do enough damage to your computer to start seeing symptoms of the damage. Now that you've done that amount of damage, you can expect to continue to see this problem, and to continue doing damage. Eventually, you'll see worse symptoms, including having your computer stop working completely.

This is very typical of this kind of damage. Most people whose computers are inadequate can run the game for some time before things bad things start showing up.

I can never understand why people are surprised that age is a factor with regard to how well hardware performs. As things get older (and/or perhaps more damaged, as may be the case here), performance will decline. Isn't that a given?

As for lag, I suspect that has more to do with the processor than with the graphics card. (I'm by no means sure of that, but it seems plausible.) It's not uncommon for laptops these days to have tons of RAM and a screaming processor that works fabulously, yet the grahpics hardware is still low-end.

I have also noticed worse performance in games other than Sims 2, lately. So I guess it would be best if I bought a new graphics card soon. My computer isn't a laptop, but I still wouldn't have expected things to get bad already, after buying it late 2009.

Late '09 is pushing four years, and that could well be "old" for a graphics card. Good news, really, because it explains the situation nicely and, since it's not a laptop, is easily fixed with a new card.